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THE PEGASUS

PROGRAMMING

MANUAL

Frontispiece: The Pegasus 2 Computer at the London Computer Centre of Perrantl Ltd.

THE PEGASUS

PROGRAMMING

MANUAL

By
G. E. FELTON, M.A.

PUBLISHED 1962

FERRANTI LTD.

THE LONDON COMPUTER CENTRE

68 NEWMAN STREET

LONDON W.I

PRINTED BY SPECIALISED PRINTING SERVICES LTD.
30-34 LANGHAM STREET
LONDON W.I

THE PEGASUS PROGRAMMING MANUAL
PREFACE

Digital computers are being put to more and more diverse uses and their numbers are rapidly
increasing. The programmer has a key role to play in this technological explosion. He and his
fellows are needed in ever greater numbers since the machines are useless without their skill.

This handbook describes the programming techniques used with Pegasus, a medium-sized computer
of which about forty are being used in business and commercial data-processing, in technical and
scientific calculations and in educational work. It is hoped that many of these techniques will be
of use also to those concerned with programming and applying other machines of similar scope.

This manual originated in a series of programming courses for Pegasus which started in October,
1955, for which an earlier manual (Perranti document CS 50) was written by Mr. P.M. Hunt and the
author. The need for a more comprehensive manual, including descriptions of the facilities provided
in Pegasus 2, has led to the present volume. While attendance at one of these courses is desirable
for anyone wishing to prepare programmes for Pegasus, an attempt has been made to keep this manual
self-contained except for a set of programming exercises, which is available separately from the
publishers (Perranti document CS 204). At the same time it should be stressed that no one can expect
to master a subject such as this one without writing programmes and running them on a computer.

This book was originally written for Pegasus 1. The author is grateful to Mrs. Mary Blyton, who
adapted it to include Pegasus 2 and collected the material for most of the Appendices, as well as pre-
paring the manuscript for printing. He is also indebted to Mr. M.J. Marcotty, who wrote the whole of
Chapter 11 and advised on the additional material relating to Pegasus 2. The author would also like to
thank many of his other colleagues, too numerous to name, with whom he has had many fruitful discussions.
The editing and printing of a book of this kind raise many problems; the author is grateful to the
following, who have contributed much to their solution: Mr. J. W. Moffatt, Mr. J. P. Nicholson, who has
also compiled the Index, and Mr. H. G. Stanton (of Specialised Printing Services Ltd.), whose care and
helpful advice have been invaluable. Special thanks are due to Mr. B. B. Swann, whose encouragement and
practical support have made this book possible. Lastly the author wishes to thank his wife, who
loyally endured much interference with their home life while he was writing the book.

G.E.P.

CONTENTS

Page

CHAPTER 1

INTRODUCTION - DIGITAL COMPUTERS

1. 1 Computers . .

1. Z Automatic digital computers . .

1.3 Programming

1.4 Comparison with desk calculators

1.5 Flow-diagrams

1.6 Numbers

1.7 Words

1. 8 Words representing numbers . .

CHAPTER 2

PEGASUS

2. 1 A Pegasus Installation

2. 2 ITie main store . .

2. 3 Hie computing store

2.4 Outline of operation ..

2.5 The written form of an order

2.6 The internal form of an order

2.7 Hie orders of groups and 1

2.8 Writing the programme

2. 9 The special registers . .

2. 10 Tlie orders of group 4 . .

2. 11 Overflow

13
13
15
17
17
18
20
22
23
24
25

CHAPTER 3 THE ORDER-CODE

3. 1 Multiplication . .

3.2 Rounded multiplication

3.3 Cumulative multiplication

3.4 Double- length addition and subtraction

3. 5 Division . .

3.6 Rounded division

3.7 Shifts, the orders of group 5

3.8 Jumps, the orders of group 6
3. 9 Stopping the computer . .

3. 10 Main-store transfers, the orders of group

3. 11 Logical operations

3.12 Orders in binary, pseudo order-pairs

26
29
30
31
32
34
36
39
44
46
50
55

CHAPTER 4 SOME SIMPLE PROGRAMMES

4. 1 Outline of output

4.2 Subroutines and the organisation of a programme

4. 3 Putting a prograirane into the computer

4.4 A simple complete programme - "Special Factorlze"

4. 5 Another complete programme . .

4.6 Relative addresses

58
62
67
70
74
79

CHAPTER 5 MODIFICATION

5.1 Modification and counting

5.2 Modification of the arithmetical orders ..

5.3 Modification of the block- transfer orders; the unit-modify order

5.4 Modification of the single-word transfer orders

5.5 Setting modifiers and counters

82
84
87
89

CONTENTS

CHAPTER 5 MODIFICATION (Contd. )

5.6 Some standard loops of orders

5, 7 Some special loops

5.8 Modification of other orders

5.9 A complete programme using modification

5. 10 Programming tricks

Page

107

ill

116

CHAPTER 6

INPUT AND OUTPUT

6. 1 Punched paper tape

6.2 Output

6.3 Input

6.4 The tape-editing equipment ..

6.5 The preparation of programme tapes

6.6 The design of subroutines for input and output

6.7 The monitors and control panels

121
122
124
126
128
133
136

CHAPTER 7 THE INITIAL ORDERS

7. 1 General description

7.2 The use of the Initial Orders by a programme

7.3 The detection and correction of blunders

7.4 Summary of the directives on tape ..

7.5 Manual directives

7.6 Block-transfer punching

7. 7 Binary input and output

7.8 Detailed description of the Initial Orders

142
144
145
151
153
156
158
160

CHAPTER 8

ASSEMBLY

1 The purpose of Assembly

2 Cues and tags

3 Cue-lists and programmer' s subroutines . .

4 Preset -parameters

5 The preparation of a programme for use with Assembly

6 The Library

7 The magnetic tape library

8 The C- directive

9 Detailed description of Assembly ..

166
167
171
174

178
182
187
188
189

CHAPTER 9 SOME PROGRAMMING TECHNIQUES

9. 1 Floating-point operations

9. 2 Interpretive and conversion programmes

9. 3 The Autocode

9.4 The Matrix Interpretive Scheme

9. 5 Double-length floating-point arithmetic

192
198
200
209
219

CHAPTER 10 MAGNETIC TAPE

10. 1 General description of magnetic tape equipment

10.2 Programming with magnetic tape equipment

10.3 Dealing with magnetic tape failures

10.4 Magnetic tape prograimies In the Initial Orders

227
227
239
241

CHAPTER 11 PUNCHED CARDS AND LINE PRINTER

Pegasus 2 punched card system

Card usage

Handling six-bit characters . .

Transfers of data to and from buffer stores

Code CdMiiversion and the Code table

Loading the card control buffers . .

Programming of punched card operations

Pseudo off-line working

Line Printer

AJS.

the 76 -order

243
244
248
250
251
256
262
266
268

CONTENTS

APPENDICES

1. The Pegasus Order- Code

2. The Lesser Library

3. Special Register 53 - Creed 3000 Punch

4. A guide to the timing of programmes

5. Abbreviations and Symbols

6. Entries to Initial Orders Routines

7. Index to the Library . .

Page

271

275
277
278
281
283
285

INDEX

309

SUMMARISED PROGRAMMING INFORMATION

319

PLATES

Frontispiece:

Plate
Plate
Plate
Plate
Plate
Plate
Plate
Plate
Plate
Plate 10
Plate 11
Plate 12
Plate 13
Plate 14
Plate 15
Plate 16

The Pegasus 2 Computer at the London Computer Centre of

Ferrantl Ltd.

A general view of Pegasus 2

The control desk of Pegasus 1

The paper tape input equipment

The paper tape output equipment of Pegasus 2

A tape box type A4 with punched paper tape

A hand spooler type A13

The full set of tape-editing equipment

The simplified set of tape-editing equipment

A keyboard perforator

The keyboard of a Creed Model 75 teleprinter

The programmers' control panel of Pegasus 2

The monitor panel of Pegasus 2

A unipunch in use for tape splicing

The controls of an ElectroData magnetic tape mechanism

ElectroData magnetic tape mechanisms attached to Pegasus 2

A card reader (left) and punch (right) attached to Pegasus 2

Note:

Some portions of the text can be omitted at a first reading without loss of
continuity. Such portions are marked in the left-hand margin by ▼ at the
beginning and ▲ at the end.

Chapter i

Introduction — Digital Computers

In this Chapter we describe digital computers in general and introduce some of the basic ideas of
programming and the way numbers are represented in a computer.

1. 1 Computers

This book is concerned with automatic electronic digital computers. A digital computer is a machine
which can perform arithmetical operations on numbers represented in digital form. This way of
representing numbers is the one with which we are most familiar, since it is in everyday use. For
example the number 53 might be represented (in a mechanical digital machine) by two gear wheels, each
with 10 teeth, one turned through 5 teeth and the other 3 teeth, relative to some standard position.
In an electronic machine this number might be represented by two trains of pulses containing 5 and 3
pulses respectively. The ordinary way of writing numbers is digital.

By contrast, in analogue computers numbers are represented by some physical quantity, such as
length, angle or electrical potential. In an analogue machine arithmetical operations are performed
by using some law of Physics, e.g. Ohm's law, and then making a measurement to find the answer. The
most familiar analogue computing device is a slide rule, which uses lengths to represent numbers (the
lengths are proportional to the logarithms of the numbers they represent); these lengths are added and
subtracted mechanically to give lengths corresponding to products and quotients. In an electronic
analogue computer numbers are usually represented by electrical potentials and the machine contains
circuits for producing potentials proportional to sums, products and so on; these potentials can be
measured to provide the results.

The precision of an analogue machine is limited by the precision with which the physical quantity
used can be measured; it is seldom greater than two or three decimal figures. To increase this
precision may be very difficult, and certainly expensive. In a digital machine the precision can be
increased as much as desired simply by allowing enough digits in the numbers; this is usually quite
easy at the time the design of the machine is being laid down. Most digital computers use numbers
having from 8 to 12 decimal digits; this may seem over-generous since the raw data of a problem may
be given to only three digits and this may be enough in the results. But it should be remembered that
a computer may perform thousands of operations before arriving at these results, and rounding errors may
therefore build up alarmingly. Further, the starting numbers and the intermediate quantities and
results may vary over a very wide range.

This is not the place for a comparison of analogue and digital machines; it is enough to say that
each has its uses. An analogue machine will perform certain restricted operations with great speed and
efficiency but it cannot have the range and flexibility of a digital computer.

Probably the most familiar of digital machines is the ordinary desk calculator. This is a
mechanical or electro-mechanical device; it is not fully automatic since it has to be individually set
up for each arithmetical operation. Some of these machines can divide or evaluate a square root at a
single setting but these are automatic only in a very limited sense.

1.2 Automatic digital computers

Automatic digital computers originated in the work of Charles Babbage in the early nineteenth
century. Babbage proposed a digital machine capable of doing extended calculations without human
intervention. This machine was, of course, to be purely mechanical. Its inventor's ideas seem to have
been ahead of the technical possibilities of his day and it was never built. The first automatic
digital computer to be made was the Automatic Sequence Controlled Calculator, an electro-mechanical
machine which was not finished until 1944. Since this date the situation has been transformed by the
speed, flexibility and reliability of electronics.

An automatic digital computer is a digital machine which can perform a large number of arithmetical
operations when once set up. Generally such a machine will have a single arithmetical unit which is
used repeatedly to do different operations on various numbers. This unit is sometimes called the mill,
a term originated by Babbage. Since there is usually only one mill there must be arrangements for
storing numbers, for selecting them and passing them to the mill, and for storing the results produced
by the mill (since these may be required again at a later stage).

An automatic digital computer of this kind is performing only one operation at any given moment;
after completing one operation it proceeds to do another, and in general it will perform a long sequence
of operations with great rapidity. The machine must therefore have a store of some sort in which can be
placed orders or instructions (the term command is sometimes used); the control unit of the machine
extracts these orders one by one from its order-store and obeys them. As a rule a single order will
cause the computer to carry out a single operation, e.g. adding two numbers together or moving a number

- 1

1.3

INTRODUCTION TO DIGITAL COMPUTERS

from one part of the number- store to another. In most digital computers the orders are expressed in a
numerical code and they can therefore be stored in the same store as the numbers; in general a part of
this store will hold numbers and another part orders. The store is sometimes referred to as the memory
of the machine t.

The store of a digital computer must be capable of "remembering" numbers and orders until they are
required. It must be able to give up any item of stored information and to record new information in
place of the old. The use of electronics in the mill means that a pair of numbers can be added in
considerably less than a millisecond (a thousandth of a second, often abbreviated to millisec). If this
speed is not to be wasted the storage devices must also operate at high speed. These requirements have
in the past been a major source of difficulty in the design of computers and developments in storage
have as a rule lagged behind those in other parts of the computer.

A computer must be able to communicate with the rest of the world. We must be able to "tell" the
machine what operations it is to perform, to supply it with the numbers on which to operate, and to
extract from it the results of its work. The machine must therefore have input and output devices, and
these must be fast and reliable in operation.

We see, therefore, that there are the following main parts of a digital computer:

(a) a store for holding numbers and orders,

(b) a control unit which can extract orders from the store, interpret them, and direct the

operations of the rest of the machine,

results back to the store,

(d) input equipment (e.g. punched card or paper tape reader),

(e) output equipment (e.g. printer or card or tape punch).

These parts and their interconnections are shown in Fig. 1. 1. It will be seen that the input and output
devices are shown as connected to the mill; this is usually the most convenient arrangement. The
connection labelled "discrimination" is described below.

STORE

orders

discrimination

-► Information paths
-► Control paths

Pig. 1.1 Block-diagram of a Digital Computer

1 ^ Pi*ofifi*flTtHni yiff

Let us consider how a digital computer is applied to the solution of a problem. The first essential

step is to formulate the whole problem precisely and unambiguously; if the problem is a scientific one

this formulation should preferably be in mathematical terras, and an equivalent degree of precision

should be aimed at in other kinds of problem. This step is usually the hardest part of the whole

process and the one where skill and experience are most important; it may well require considerable

time and study. We shall not discuss this subject further here since it belongs properly to the field

of study which gave rise to the problem.

The next step is the examination of this precise formulation of the problem in order to find the
method of solution best adapted to the available computing tools. This step may, in scientific
problems, require a knowledge of Numerical Analysis; and it may In any case be combined with the
previous step. This subject also is outside the scope of this book, but there is one point which must
be made in this connection. A digital computer can, at least in principle, undertake any task which
can be expressed precisely as a finite sequence of finite arithmetical operations. It must be
emphasised that it is only problems of this sort which can be handled by a computer; but this is not
such a severe restriction as might appear at first sight and there are many non-numerical problems
which can be reduced to arithmetic; for example, much work is now being done on the automatic transla-
tion of languages by computers. The practical limits on the range of problems which can be dealt with
are largely Imposed by the storage capacity and speed of the computer and its ancilliary equipment.

We shall assume, therefore, that we are presented with a definite numerical process, which we must
express in a form assimilable by the computer. That is, we have to programme^^ the problem, or prepare
the programme of orders for the computer. In some circumstances it may be useful to subdivide this
process into two stages. In the first stage, programming in a narrower sense of the word, the numerical
process is converted into a flow-diagram, or flow- chart, showing what steps the machine is to take and
how one operation leads to another. In the second stage, which is often called coding, the actual
orders which the computer will have to obey are written down in some convenient code; this is
relatively simple, once the flow-diagram has been prepared. The exact way in which programming and
coding are distinguished (if at all) depends very much on individual preference and no inviolable
rules can be laid down - except that great care is needed at all stages. The next step is to punch out
the programme, order by order, on paper tape or cards and to feed the result into the computer, along

+ Such anthropomorphisiBS are perhaps to be deprecated, but they sometimes provide the simplest way of

describing the operations of the computer.
"t^t The spelling program is sometimes used.

2 -

INTRODUCTION TO DIGITAL COMPUTERS 1.4

with the numerical data (though these may be fed in as a separate operation). At this stage the orders
and numbers appearing on the tape or cards are simply read by the computer, converted into its own
internal code, and placed in the store. When this input process is complete the store of the machine
will contain the whole of the programme and some or all of the numerical data; at this point this
computer starts to select and obey the orders of the programme one by one. Some of these orders will
ultimately cause the machine to print or punch the results of the calculation.

It will be clear that the preparation of the programme for a particular calculation may well be
time-consuming. When it has been written the programme will have to be checked and got working on the
computer; this stage is usually called the development of the programme. When the preparation and
development are complete the programme may be used repeatedly with different numerical data. This
means that it may be uneconomic to write a programme for a calculation which has to be done once only;
but if a calculation has to be done many times, perhaps at regular intervals, the cost of preparing the
programme can be distributed and it will usually be found that a digital computer offers by far the
cheapest method of doing the work.

The orders which the computer obeys in the course of executing a programme are selected from the
set of available types of order built into the machine. This set of orders is called the order- code of
the computer; it is important for the programmer that a comprehensive set of orders should be available,
and that the exact effects of each of them should be known to him. It is very helpful if the orders
are systematically arranged and are free from objectionable exceptions and omissions. Many of the
orders in the order-code of any computer are concerned with simple arithmetical operations; but there
are others (for example, those for transferring blocks of numbers from one part of the store to another)
which are needed only because the machine is automatic.

The orders In many computers are normally obeyed sequentially; that is to say, they are extracted
one after another from adjacent places in the store and are obeyed. Certain orders, called jump orders,
may break this regular sequence and cause the machine to start selecting its orders from some other
specified place in the store; whether this jump occurs may be conditional on, for example, the sign of
some numbers. In some computers the jump orders are called control-transfer orders since they can be
said to transfer control to some other part of the programme. Other names are test or discrimination
orders. All automatic digital computers have jump orders of some kind and they add enormously to the
flexibility of the machine. When a conditional jump order is obeyed certain information is passed from
the mill into the control unit; this information is used to determine whether or not the jump occurs
(the information passes along the path labelled "discrimination" in Pig. 1.1).

Nearly all calculations are, at some stage or other, highly repetitive and the programmer can take
advantage of this by writing groups of orders and arranging (with the aid of jump orders) that the
machine obeys the orders in each group several times. This is of great importance, as will shortly
become evident.

A book devoted to programming loses realism unless it is related to a specific machine. In this
book, the operations are described in terms of the Perranti Pegasus Computer. They are, however,
readily adaptable to other digital computers and the reader who masters the techniques described in the
pages which follow can approach any modern computer with confidence.

1.4 Comparison with desk calculators

An analogy with other methods of computing may be useful here. Let us suppose we have a desk
calculating machine equipped with an unintelligent, though extremely reliable, operator and we have some
particular numerical problem to be solved. A suitable numerical process for arriving at the solution
must be found and this must be written down as a series of simple operations and given to the operator
as a precise sequence of orders or instructions. Provided this work has been done correctly and the
operator follows slavishly the instructions he has been given the solution of the problem will eventually
be obtained. It should be noted that the operator need not understand the reasoning which led to his
instruction list nor the significance of his operations. Purthermore the same sat of instructions can
be used again on another occasion to solve a similar problem involving different numerical data. It is
usually advisable to include a few checks in the calculations, and this can be done by giving extra
instructions to the operator.

The procedure for solving a problem with the aid of an automatic digital computer is similar;
the operator is replaced by the control unit of the computer, his desk calculating machine becomes the
mill, and his list of instructions becomes the programme of orders for the machine to obey. The
work-sheet on which the operator writes his intermediate numbers can be likened to the store of the
computer.

It will be seen that the whole process is in principle very much the same. The main practical
points of difference arise from the fact that most desk calculator operators have a fair amount of
intelligence (they will not, for example, attempt to divide a number by zero as some automatic
machines may). A digital computer may well be 10,000 times as fast as the operator with his desk
machine so that the difference of speed is of great importance. There are two main points of
consequence resulting from this enormous speed ratio; first, problems can now be tackled efficiently
which have hitherto been beyond the reach of computation or have been otherwise uneconomic; and
second, a numerical process might well be chosen which would not be used on a desk calculator. In
fact the digital computer is so much faster than a desk calculator as to make a qualitative, rather
than quantitative, difference.

1.5 Flow-diagrams

We shall now illustrate some of the points mentioned above with the aid of flow-diagrams. In
these diagrams each "box" represents a simple operation or a group of such operations.

(a) Let us first suppose we have in the store of the machine a list of whole numbers, each of which
may have any value from l to 100, except that the last number in the list is known to be zero. Suppose
that the computer is equipped with a printer as an output device and we wish to use this to print all
those numbers in the list which are not less than 50. The flow-diagram of Pig. 1.2 shows a possible
sequence of operations.

3 -

1.5

INTRODUCTION TO DIGITAL COMPUTERS

I START j-

Select the first
number in the list

i__jk

Test if zero

yes

Subtract 50

Test sign

negative

positive or zero

Add 50

Select the next
number in the list

Fig. 1.2 Flow-diagram of a programme to print those

numbers in a list which are not less than 50.

It will be noted that the process consists simply of exa' ining each number in turn; if the
number is zero we know we have reached the end of the list, and if it is not we arrange to avoid the
"print" operation if the number selected is less than 50, The importance of the test or conditional
jump operations is clear.

(b) In the next example, we shall assume that an input device is provided which can be used to read
numbers into the computer one at a time. These numbers might, for example, be punched on paper tape or
cards. Suppose that there are 1000 positive numbers waiting to be read and we wish to print the
largest of them. It is convenient to use an algebraic notation; the letter x denotes one of the
numbers read in from the input device and y stands for the largest number so far read. We read in

each nuraiber in turn and compare it with y, if it is less than y or equal to it we simply pass on to the
next number. If, however, the number is greater than y we must increase the value of y to the new
number before reading in the next one. We also have to arrange to count the numbers as they are read
in (a quantity c is used for this) so that we know when we have finished, at which point y is to be
printed since it is now the largest of all the numbers. The flow-diagram is shown in Figure 1.3.

The technique shown in this flow-diagram for counting should be particularly noted as it occurs
frequently. The main part of the above programme consists of a loop or cycle of orders which the
computer is to obey exactly 1000 times; after each repetition a counter (in this case c) is reduced by
unity until, after the orders of the cycle have been obeyed 1000 times, it is reduced to zero and the
computer passes on to the next part of the programme. The details of this counting process depend very
much on the facilities available in a particular computer; it may, for example, be more convenient on
some machines to start with a negative counter and to increase it by unity at each repetition until it is
no longer negative.

(c) As a further example let us find the smallest prime factor of a positive whole number N, i.e. the
least number (other than 1) which divides exactly into N. We shall assume that N is greater than 1. A
process which may be used is first to test if A^ is even, in which case the answer is 2; if N is not

even we try dividing it by 3,5,7,9 (i.e. by consecutive odd numbers) until we find a divisor.

Strictly speaking we need try only prime divisors but it is simpler to include composite numbers, such
as 9, than to omit them, even though we know when we reach one that it cannot divide N exactly (for if

N is not divisible by 3 then it is not divisible by 9). If iV is a prime the first divisor we shall find
will be N itself, but there is no need to go as far as this. As the trial divisor (d. say) steadily
increases, the quotient (g, say) will steadily decrease; in the absence of a successful trial d will
ultimately exceed q. Note that if rf is a factor of N then so also is q, so that each trial with d is in
effect also a trial with g. It is consequently pointless to go beyond the stage at which d becomes
equal to or greater than q, for any quotient obtained after this must be one of the numbers already tried
unsuccessfully as a divisor (or else the quotient is even, and we know N cannot have an even divisor).
We need therefore a cycle of orders which tests whether our trial divisor d divides N; if it does not
we increase rf by 2 and re-enter the cycle provided d is less than the quotient q. The flow-diagram of
Pig. 1.4 (page 6) shows the process. Note that it will in fact work even if yv = 1.

INTRODUCTION TO DIGITAL COMPUTERS

1.5

START

Set c = 1000

Set y =

Read in one
number, x

Form Difference
y - jc

Test sign

negative

Replace y
by X

Replace c
by c - 1

Test if new value
of c is zero

yes

Print y

positive

or zero

STOP

Pig. 1. 3 Flow-diagram of a programme to print the largest
of 1000 positive numbers read in one by one,

(d) Another Illustration is provided by a programme to evaluate the square root of a positive number a.
We shall use Newton's process; this is based on the fact that if x^ is an approximation to /a, then

H'.^^)

is a better approximation. This calculation can be repeated with x^ in place of x. to yield a further
approximation jcg, and so on. The successive approximations are in general connected by the relation

n + i

H'n^l).

If we are fortunate and if a is a perfect square we may ultimately arrive at the exact value x^ = /a and
the process may be said to have terminated. But as a general rule this will not happen and we shall
have to be content with an approximation; for example we could stop whenever two successive approxima-
tions differ by not more than some small preassigned quantity h.

As an illustration let us obtain an approximation to /3, We may take x^ = 1 as a first guess; then

-' f('>^t) = i('*T)

and

T = l-'^S'

= 1(2 ^ f )

iCi^ yfu) = fe = 1.7321428

The correct value to 6 decimal places is 1.732051. It will be seen that this process converges very
rapidly - it is in fact an example of what is known as a "second-order" iterative process, in which the
number of significant figures Is approximately doubled at each step.

5 -

1.6

INTRODUCTION TO DIGITAL COMPUTERS

Test if N
is even

yes

♦ Set d = 2

Set rf = 1

P Increase d by 2

Divide N by d;
quotient = q
remainder = r

Test if r =

yes

Evaluate d - q

negative

Test sign

positive

Set d = N

Pig. 1.4 Flow- diagram of a programme to find the
smallest prime factor of a number N

Returning to the general process for /a, the difference between two successive approximations is

n+l

('»

say.

We can therefore readily compute this quantity from the value of x^ and examine it. If it is greater
than h in absolute value we simply add it to x^ to get x^^^ and then repeat the process; if it is less
than h (or equal to it) we have finished.

Any of these flow-diagrams forms a suitable basis for a programme or a part of a programme. It will
be remarked how each process has been broken down into very simple steps and how each step leads
unambiguously to the next. Programming has been described as "explaining the problem to the computer
in words of one micro-syllable".

The notation used for these flow-diagrams has been chosen to be self-explanatory. If the prepara-
tion of flow-diagrams is to be undertaken systematically as a preliminary to the programming or coding
of a process it is probably advisable to use a more rigid system to exclude possible ambiguities.

1. 6 Numbers

The ordinary way of writing numbers may be called a decimal (or denary) system, since it is based
on the number 10, which is called the radix of the system. For example the number written 5428 is an
integer (or whole number) whose value is

5428 = (5 X 10^) + (4 X 10^) + (2 x 10^)+ 8 ,

which may alternatively be written

(^([(5 X 10) +4] X 10} + 2) X 10 + 8.

In this number 5 is the most-significant (or left-most) digit and 8 is the least'significant (or right-
most) digit. The contribution made by each digit to the value of the number is just the value of the
digit multiplied by a power of 10 determined by the position of the digit in the written form of the

6 -

INTRODUCTION TO DIGITAL COMPUTERS

1.6

START

egative

Set jc = 1

First approximation

Evaluate —

Subtract x
to get - - X

Halve this,

(7- ^)= y «^y

Replace x by

^ + y

Compute
> improved
approximation

Test sign of y

negative

positive

or zero

Change sign of y

Add h

Test sign of result

positive or zero

Repeat cycle
if difference
is h or more

Pig. 1.5 A process for evaluating v^a; the result is x.

number. Fractions and mixed numbers may be expressed by writing digits to the right of a decimal point,
for example

which may be written^

27.93 = (2 X 10) + 7 + (9 X A) + (3 X i g) .

(2 X loS + (7 X 10°) + (9 X 10" S + (3 X lO"^).

The decimal point is not usually written in integers, it is understood to lie to the right (e.g. 5428
could be written 5428. or 5428.0).

This is not of course the only way of representing numbers. Sums of money in sterling and periods
of time in hours, minutes and seconds are examples of mixed-radix systems; for example £461.16.8
represents

{(4 X 10^ + 6 X 10 + 1) X 20 + (1 X 10 + 6)} X 12 + 8 pence,

and 3 hours 24 minutes 52 seconds represents

{(3 X 60) + (2 X 10 + 4)} X 60 + (5 X 10 + 2) seconds.

t The negative exponent in expressions such as 10"^ and 2'"'^ simply means that we have to take the
reciprocal. For example 10-3 is simply another way of writing I/IO"^, and 2-"+ = 1/2** - 1/16.

7 -

1.7

INTRODUCTION TO DIGITAL COMPUTERS

Various combinations of such radices as 5, 12, 14, 16. 20, 60 are in general use for weights and
measures in many parts of the world; such systems are widely understood despite their complexity.

A simple system which finds some application in computers is the octal system. This is similar to
the ordinary decimal system but is based on the radix 8 instead of 10; thus the number- written 2736 in
the octal system has the value

(2 X 8^) + (7 X 8^) +(3x8) +6,

= (2 X 512) + (7 X 64) +(3x8) +6,

= 1024 + 448 +24+6.

= 1502

in the usual decimal system. In this system only 8 different digits are needed (0,1,2, ..,7) instead of
the usual 10. Octal fractions may be written by introducing an octal point, for example 3.5 means 3%.
This is clearly much simpler than some of the mixed- radix systems with which we are familiar.

The simplest system of all, and one which is used in most digital computers (including Pegasus) is
the binary system, which is based on the radix 2. This has the great advantage of needing only two
different digits (0 and 1) for writing any number. The number written 1101 in binary has the value

(1 X 2^) + (1 x 2^) + (0 X 2) + 1,

= 8 + 4+0 + 1,

= 13.

in the decimal system. We may write fractions and mixed numbers if we introduce a binary point, for
example

10.101 = (1 X 2) + + (1 X 2'S + (0 X 2"^) + (1 X 2"^.

= (1 X 2) + + (1 X 1) + (0 X ^) + (1 X -|).

= O + 1. + J. = 9.5.

It might appear at first sight that any departure from the usual decimal system would introduce
many grave disadvantages to the programmer and user of the computer. This is not so however; in fact
one need seldom be conscious that the machine is operating in binary. This is because all the numbers
fed into the machine and all those which come out of it are in decimal (or in some other convenient
system). All the necessary conversions to and from the binary system are merely arithmetical operations,
which the computer itself does efficiently. When considering the internal operations of the computer
it is usually enough to think of them as being carried out simply on numbers; one need not usually
consider the way in which these numbers are represented inside the machine.

Table 1. 1 shows several numbers in binary together with their decimal equivalents. It will be
noticed that integers (whole numbers) have more digits when represented in binary than in decimal; as
a rough rule 10 binary digits are about equivalent to 3 decimal digits t. The word bit is often used as
an abbreviation of binary digit.

It is, of course, essential for the computer to be able to use negative numbers as well as
positive ones. The ways in which negative numbers are usually represented will be described later.

It should be noted that the octal system described above is closely related to the binary system.
In fact, since 8=2^, a single octal digit is exactly equivalent to three binary digits. An octal
number can be converted to binary by simply writing down the binary equivalent of each of its digits,
thus 473 in octal is 100 111 Oil in binary. Conversely, the binary number

1011. 101

is 13.5 in octal (or 11% in decimal). Note that one cannot use such a simple process for converting
from binary to decimal, or vice versa. It is sometimes more convenient to write binary numbers in
octal form since the conversion is easy and the result is more compact and more easily remembered.

While it is as well for the programmer of a binary computer to be familiar with the elements of
binary arithmetic, fluency in handling binary numbers is not at all necessary. One point worth noting
is that if we move the binary point one place to the right in a binary number we shall effectively
double the value of the number, e. g.

10. 101 is 2|

and 101.01 is 5^.

In the decimal system such an operation multiplies the number by 10 of course. In the same way if we
move the binary point one place to the left we shall halve the value of the number.

1. 7 Words

The numbers normally handled in a computing machine contain a certain fixed number of digits. It
is convenient to use the term word for such a number. A word is a group of digits normally handled
together by the computer. Words may be used to represent either numbers or orders, and this is the
chief reason for introducing the term. In most computers the words representing numbers and those

t This is because 2 = 1024 O:' 10^; a more exact ratio is Iog2l0, which is about 3.3.

INTRODUCTION TO

DIGITAL COMPUTERS

Binary Decimal

Binary

Decimal

1 or 1.0 1

0,1

1
0.5 =2"

10 or 10.00 2

0.01

0.25 =4

11 3

0.11

0.75 =1

100 4

0.001

0.125 = i

101 5

0.101

0.625 - 1

110 6

0.0101

0.3125= j|

111 7

1.01

1.25 = li

1000 8

1.1

1.5 =li

1001 9

10.1

2.5 = 2i

1010 10

110.1

6. 5 = 6|

1011 11

101.001

5. 125 = 54

10001 17

11001 25

100001 33

1100100 100

1.8

Table 1.1 .Some binary numbers with their decimal equivalents

representing orders are indistinguishable in appearance, each is merely a string of digits. Words of
these two kinds are usually stored in the same store, but the programmer often allocates certain
parts of the store to numbers and other parts to orders.

In binary computers, words of 32 to 40 or more binary digits are generally used; the word-length
is usually fixed by the construction of the machine. Pegasus has a word-length of 39 binary digits,
which is equivalent to rather more than 11 decimal digits. In most such computers a word is
represented by a train of pulses of electrical potential; the presence of a pulse Indicates a "1"
digit and its absence a "0" digit. The pulses representing a word follow one another at an interval
called the digit- time; in Pegasus this is 3 microsecondst. Digit-times of 1 to 30 microsec are in
general use. The digit- time is fundamental to the design of the computer; it is derived from a dock
waveform, which has a frequency of about 333000 cycles per second (i.e. 333 kilocycles per second) in
Pegasus. The operation of the whole computer is synchronised by the clock waveform. Computers of this
kind are called serial machines, since the pulses representing a word may be sent serially along a
single channel. This is in contrast to parallel machines in which, for example, 39 channels would be
used to transmit simultaneously all the pulses of a 39-bit word. It is obvious that a parallel machine
will as a rule be faster than a serial one and will contain very much more equipment. Some computers
strike a compromise by operating in a mixed mode, known as series'parallel, in which, for example, one
might use three channels, each carrying a train of 13 pulses. All small and medium-sized computers
operate in the serial or series-parallel modes; some large machines work in the parallel mode.
Pegasus is a medium- sized serial computer.

In a serial computer the time needed for the transfer of a word from one part of the machine to
another is called a word-time (the term beat is also used). In Pegasus a word-time is equal to 42
digit- times (i.e. 126 microsec) since there are three unused gap digits between the end of one. 39-bit
word and the start of the next. The duration of any operation in the computer is always an integral
number of word- times. When finding out how long some particular programme is going to take on the
computer, we usually count the number of word- times and finally convert the total into seconds or
minutes.

1.8 Words representing numbers

A word may be used to represent a number, in fact this is one of the major uses for words. If the
digits of the word are written out side by side in the usual way (with the most-significant digit on
the left) we shall get an integer (or whole number). This is one way of interpreting the word, or
assigning a value to it, and we shall call it the integer convention. There is an implied binary point
just to the tight of the least-significant digit.

In Pegasus, as in most binary computers, the left-hand (or most-significant) tt digit in a number-
word is used to indicate the sign of the number; it is called the sign-digit, (or sign-bit) and is in
a positive number (or zero) and 1 in a negative number. We shall explain shortly how negative numbers

t A microsecond isamilllonth of asecond; it is usually written microsec or, sometimes, /i.aec.

tt The abbreviations ms and Is will often be used with the meanings most (or more) significant and least
(or less) significant, respectively. When we write words out it Is always understood that the ms
digit is on the left, as is customary when writing ordinary decimal numbers.

9 -

1.8 INTRODUCTION TO DIGITAL COMPUTERS

are represented by words. The digits of a word are numbered fi-om left to right for reference purposes;
the sign-digit is digit 0, the digit to the right of it is digit 1, and so on. The last or Is digit is
digit 38. We shall often write D in front of the digit-number so that, for example, D12 means digit 12.

Instead of using the integer convention we could insert a point (a binary point in Pegasus of
course) between two specified digits of the word. This point is not represented in the word and its
position is largely a matter of convention. In Pegasus the binary point is normally placed between DO
and Dl, i.e. immediately to the right of the sign-digit, so that a typical 39-bit word may be written

0. 10001 10111 11001 00010 10000 01001 10001 001

which is a binary fraction. This way of assigning a value to the word will be called the fractional
convention.

The word

0.00000 00000 00000 00000 00000 00000 00000 000

represents the number zero on either the fractional or the Integer convention. The smallest positive
number is represented by the word

0.00000 00000 00000 00000 00000 00000 00000 001

whose value is 1 on the integer convention, and 1/2"^^ (or 2"^^) on the fractional convention (this is
about 0.00000 00000 036 in decimal). The largest positive number is represented by the word

0.11111 mil mil mil imi iim iiiii m

On the fractional convention this has a value Just less than unity, in fact 1-2"^° (0.99999 99999 964
approximately in decimal). On the integer convention it can be shown that its value is
238 _i = 27 48779 06943, i.e. rather more than 250 000 million. Thus, a word representing a non-
negative nurabert can take values from to 27 48779 06943 on the Integer convention, or from to 1-2"^^
in steps of 2~^^ on the fractional convention. These numbers are therefore expressed by the equivalent
of rather more than 11 decimal digits.
ff Y The value of a non-negative word can be found by adding up contributions from each of its digits

which are 1' s. On the fractional convention digit Dl contributes ^ if it is a 1; Z)2 contributes % and
so on; in general digit k contributes 2"^ to the numerical value of the word if it is a 1. On the
A, integer convention each digit contributes 2^^ times as much.

Let us now consider negative numbers. If we use a desk calculator which handles numbers of 10
decimal digits and we subtract 48 from we shall get the result 99999 99952. This may be regarded as
an alternative way of writing the negative number -48. If negative numbers are written in this way
they may be added and subtracted correctly on the calculator, provided the results are consistently
interpreted. We say that negative numbers written in this way are represented in complementary form.
The complementary form of a negative number may be obtained by subtracting the absolute (unsigned)
value (in this case 48) from 10^°, which is the number lying Just outside the capacity of the
calculator. The usual way of writing signed numbers Is by means of a sign (+ or -) and the modulus*
of the number.

Either of these ways of representing signed numbers may be used in a digital computer; and other
systems are occasionally employed. If a computer uses the sign and modulus representation there may be
two different ways of representing zero (i.e. as +0 and -0); it is important that these should be
treated in the same way.

Pegasus uses the complementary system for negative numbers and we shall now describe this in more
detail. The complements are taken with respect to 2^^, which plays a role corresponding to 10 ^"^ in the
above example of a decimal desk calculator. Consequently a negative number which is small in absolute
value is represented by a word having a string of 1' s on the left (corresponding to the string of 9' s
above). For example, the integer 44 is represented by the word**

0,00000 00000 00000 00000 00000 00000 00101 100

We can find the word representing -44 by subtracting the above word from 0. The result is

1.11111 11111 mil mil mil mii iioio loo

One way of determining the value of a negative number-word is to subtract it from and evaluate the
resulting positive word according to the usual rules. The process of subtracting a number from is
usually called negating the number, or changing its sign. It should be particularly noted that this
is not done by simply changing the sign-digit (as it would be if the sign and modulus representation
were used). An easy way of changing the sign of a given word on paper is to start at its Is digit (on
the right) and, proceeding to the left, to copy all the 0' s (if any) until we find a 1; this 1 is also
copied. The remaining digits are then reversed, i.e. we write 1 for 0, and for 1. Tables 1.2 and
1.3 illustrate how numbers are represented by words in the two important conventions.

t A non-negative number is a number which is either positive or zero. Zero is regarded as being neither
positive nor negative.

tt T

i The text between these symbols may be omitted at a first reading.

* The modulus of a number is simply its magnitude or absolute value. The modulus of 48 is 48; the
modulus of -48 is 48. Moduli are conventionally indicated by vertical lines on each side of the
number, thus |48| means "the modulus of 48" and we can write

i3| = 3, \lVi\ = Vk. 1-21 = 2, lOl = 0.

## We shall adhere to the convention of writing the point in number-words and grouping the digits in the
way shown, regardless of the convention used to assign a value to the word.

- 10 -

INTRODUCTION TO DIGITAL COMPUTERS

1.8

It will be seen from these tables that the word whose value is 13 according to the integer
convention has the value 13 x 2"38 according to the fractional convention. The small quantity 2'^^
occurs frequently and we shall often denote it by e (the Greek letter epsilon)t; thus 13 x 2-^8
may be more conveniently written 13e. We shall refer to number-words as integers or fractions
according to the convention used to interpret them. Suppose that x^^ is the value of some word inter-
preted as a fraction, and that xj is the value of the same word as an integer; these are connected
by the equations

= 2

-f

= Xj/2

38 _

It is important to realise that xj and xp have the same digits in binary, but that their decimal
representations would be quite dissimilar.

Fraction

Word

representing the fraction

0.10000 00000

00000 00000 00000 00000 00000 000

1 _ 0-3

0.00100 00000

00000 00000 00000 00000 00000 000

_ 1
8

1.11100 00000

00000 00000 00000 00000 00000 000

e = 2-3«

0.00000 00000

00000 00000 00000 00000 00000 001

13e

= 13 X 2"38

0.00000 00000

00000 00000 00000 00000 00001 101

- 13e

1.11111 11111

mil mil mil iiiu imo on

_3_
4

0.11000 00000

00000 00000 00000 00000 00000 000

_ _3_
4

1.01000 00000

00000 00000 00000 00000 00000 000

- 1.0

1.00000 00000

00000 00000 00000 00000 00000 000

Table 1.2 39-bit words representing numbers (fractional convention)

Integer

Word representing the integer

- 1

- 13

128 = 2^

- 128

237 = 137438953472

0.00000 00000 00000 00000 00000 00000 00000 001

0.00000 00000 00000 00000 00000 00000 00001 101

1.11111 mil mil mil iim nm nm m

1.11111 mil mil mil mn mn imo on

0.00000 00000 00000 00000 00000 00000 10000 000

1.11111 mil mil mil nm nm loooo ooo

0. 10000 00000 00000 00000 00000 00000 00000 000

Table 1.3 39-bit words representing numbers (integer convention)

We can define the numerical value of a word according to the fractional convention by defining the
contributions made by its various digits (if they are 1' s) as follows:

(a) the sign- digit contributes -1,

(b) digit k contributes 2"^ (if k if not zero).

These rules may be expressed in a more compact way if we write rfj^ = or 1 according as digit fe is
or 1 (0 < fe < 38). The value of the word on the fractional convention may then be defined as

-d, +2: d,.2

-k

Its value on the integer convention is

-c/,.2^^-^2 d^.2
fe=l

3 8-Jfe

t For estimating the approximate sizes of numbers we can take e ^0.00000 00000 036.

- 11 -

1.8

INTRODUCTION TO DIGITAL COMPUTERS

A word representing a negative fraction may take values from -1.0 to -e in steps of e. It
follows that a number x can be represented on the fractional convention only ift

-1 < ;e < 1 - e,

i.e. X must be numerically less than unity, except that the value ;e = -1 is allowed. A word may
represent an integer n provided

-2^^ < n < 2^8 _ 1,

i.e.

•27 48779 06944 < n < 27 48779 06943.

A fraction x can be represented exactly by a word only if it is an integral multiple of e = 2'3 8;
in general we shall have to approximate. Any fraction can be represented with an error of at most
± ^€ by correct rounding of the last digit. Table 1.4 shows a few such approximations.

Fraction

Approximate representation by a 39-bit word

2
3

0. 10101 01010 10101 01010 10101 01010 10101 Oil

1
3

0.01010 10101 01010 10101 01010 10101 01010 101

1
5

0.00110 01100 11001 10011 00110 01100 11001 101

1
7

0.00100 10010 01001 00100 10010 01001 00100 101

1
10

0.00011 00110 01100 11001 10011 00110 01100 no

Table 1.4 Rounded approximations to fractions

If the numbers occurring in a calculation are integers or fractions they would normally be
represented according to the appropriate convention. Otherwise they must be scaled in some way. This
scaling must be done in such a way that all the intermediate quantities formed in the machine during
the course of the programme are within range; and, if fractions are used, the scaling must be such
that accuracy is not lost. This is usually possible; but when it is not, or when it is very difficult
to determine the scaling factors, then there are well-established programming techniques, such as
floating-point working, which can be used. Occasionally we may want more precision than can be got
by the normal representation of numbers, i.e. we may need more than 11 decimal digits: we can then use
double-length (or double-precision) arithmetic, in which each number is represented by two words.
These and other techniques will be described later.

If we say that a word has the value 0.75 then we obviously mean that the word is to be
interpreted on the fractional convention; and if we say its value is 94 we intend the integer
convention to be used. As a rule the fractional convention is regarded as the standard one.

The word whose value is -1 according to the fractional convention will usually be written -1.0 to
prevent confusion with the integer -1; this word has a 1 in the sign digit position (DO) only, and has
the value -2^^ on the integer convention.

Nearly all of this Section applies, with only small changes, to other binary computers using the
complementary representation (as most of them do). The fractional range -1 ^ a; < 1 applies to most
computers, though other ranges are occasionally used. The integer range depends, of course, on the
word- length.

t The following useful mathematical symbols will be employed often; they are tabulated here for the
convenience of those to whom they may be unfamiliar.

± 3 means +3 or -3 (read as "plus or minus 3"),

X = y means x is equal to y,

X f y means x is not equal to y,

X < y means x is less than y,

X ^ y means x is less than or equal to y (or x does not exceed y) ,

x > y means x is greater than y (or * exceeds y),

X ^ y means x is greater than or equal to y,

x c^ y OT X ^ y means x is approximately equal to y.

It should be noted that x < y is equivalent to saying that x ~ y is negative. For example, the
following are all true statements:

2 < 3, 2 < 3, -2 < ~1, 6 > 0, > -1. 5 ^ %, -% < -%,

We shall write, for example, -1 ^ x < 1 to mean that both -1 ^ x and jc < 1. The symbol 2 means
the sum of, for example.

2 df^.2' = d
k=l

^.2"^ + d^.2

-2

+ ^3.2"

•^ '^3e-2

-38

12 -

Chapter 2
Pegasus

This Chapter contains a brief description of a Pegasus installation and its main components. This
is followed by a description of the store of the computer and the way in which orders are obeyed and
written. A discussion of the order-code then leads into an account of the simpler orders.

2.1 A Pegasus installation

In Section 1.2 we described briefly the main parts of a typical digital computer; let us now
examine a Pegasus installation. The store in Pegasus is split into two parts; the main store, which
is large, and the computing store, which is a small, fast, working store; these will be described in
the next two sections. The control unit, which may be thought of as the "central nervous system" of
the computer, obeys orders taken from the computing store; its mode of operation will be discussed
further in Section 2.4. When it is obeying orders the control unit selects numbers from one of the
stores (generally the computing store) and makes use of the mill to perform arithmetical operations.
In Pegasus 2 the computing store, control unit and mill are housed in the main cabinet, on the front of
which are the control panels and desk (see Plates 1,2).

There are three control panels; (a) the monitor (Plate 12), on which are two cathode ray tubes
which may be switched so as to exhibit various words and waveforms; (b) the programmers' switches
(Plate 11) which are the principal controls for the computer; and (c) the engineers' switches under a
hinged flap on the desk which are not of great concern to the programmer. The control panels are
described in Section 6.7. Above the monitor panel is a clock (timepiece).

On the desk stands the basic input and output equipment comprising two tape-readers (Plate 3)
and the output punch (Plate 4). These devices handle punched paper tape, a useful medium for carrying
information (see Plate 5). Either of the tape- readers can be operated by the computer itself so as to
"read" the tape; this is the way in which programmes, numbers and other data are supplied to the
computer. The input tape which is to be read by the computer may be prepared manually by using the
tape-editing equipment, which includes a teleprinter with a keyboard and perforating device (Plates
7, 8, 9). The computer punches the results of the calculation into paper tape by using the output
punch. The output tape so obtained is usually printed out almost immediately by an interpreter
(Plate 4), which reads the holes in the tape and prints the corresponding characters on a roll of
paper. The output tape may, if desired, be used as an input tape to the computer on another occasion.

Among the programmers' switches the most important controls are the 5tarf key and the Run key.
The Start key is used mainly to cause the computer to read a programme tape, i. e. to read in and store
the orders making up a programme. The Run key can be used to stop the computer at any time or to
allow it to carry on.

We shall be concerned for most of this book with the basic Pegasus 2 computer, which has Just
been briefly described, and certain ancillary equipment. We cannot provide here a full description of
every item of ancillary equipment as the range of available equipment is continually being extended.
A Pegasus installation may include some or all of the ancillary equipment according to the nature of
the tasks it is required to perform. The following are items of ancillary equipment to which we shall
refer; further details are given in Chapters 10 and 11.

(a) Magnetic tape auxiliary storage.

(b) Punched card input and output equipment.

(d) Multiple output punches.

Although we refer specifically to Pegasus 2. much of the text which follows is relevant to both the
original Pegasus 1 with the small drum and the present Pegasus 1 with the larger drum. It will, however,
be clear from the footnotes how these versions of Pegasus differ from the Pegasus 2.

Z.Z The Main store

The first 128 words of the main store of Pegasus 2 are held on delay-lines, and the rest of the
main store is a magnetic drum revolving at about 3720 revolutions per minute +. We shall first of all
discuss the drum storage.

One revolution of the drum takes exactly 128 word-times, or roughly 16 milliseconds. tt The
surface of the drum is coated with a magnetisable iron oxide, and just clear of it are a number of
fixed heads containing coils. Parts of the drum surface may be magnetised by passing pulses through
the coils in the heads: we can therefore record a word magnetically (or write the word) by passing
the corresponding train of pulses through one of the heads. The resulting small areas of magnetisation
will occupy a 128-th part of the track swept out by the head. By virtue of a property of the iron
oxide, this magnetisation will remain until the next time this particular head is used to write on to
the same part of its track. The heads used for writing may also be used for reading the recorded

+ In Pegasus 1, the magnetic drum constitutes the whole of the main store.

tt A millisecond is a thousandth of a second; it is usually abbreviated to millisec or msec. Clearly
1 millisec = 1000 microsec.

13 -

2.2 PEGASUS

words, since the small magnetised areas will induce pulses in the heads as they are carried past by
the rotation of the drum. The reading process may be repeated as often as desired, since it does not
disturb the magnetisation of the drum.

Information recorded on the drum is, for engineering reasons, split up into channels of odd and
even digits which are recorded separately, thus, 2 heads are needed for each track of information.
There are 71 head pairs and each sweeps out a track on which 128 words may be recorded; thus, with
128 words on delay-lines, the total capacity of the main store is 9216 words, t Each of the places where
a word may be recorded is called a storage location or simply a location (the term cell is sometimes
used). In order to write a word into some particular storage location on the drum the computer has to
select the appropriate heads and then energise them at the correct time. The word may be read later
by selecting the same heads and extracting the pulses from them at the appropriate moment. Any
particular storage location is thus identified by specifying the heads or track to be selected and
the time (or angular position of the drum). The selection and timing are performed quite automatically
by electronic circuits associated with the drum; they are derived from the address of the storage
location, which can be thought of as a label permanently attached to the location and used to identify
it. The address of a storage location may be compared with the address of a house in a city, which
can be specified by means of a street name (corresponding to the track) and the number of the house in
the street; the analogy is even closer if we imagine the streets to be numbered (as in some American
cities) and we suppose there are exactly 128 houses in each street. The word stored in a particular
location is called the content of the location; it must not be confused with the address of the
location, just as one must not confuse the inhabitants of a house with its address.

The storage locations are grouped into blocks, each consisting of 8 locations; there are
consequently 16 blocks round each of the 71 tracks and 16 blocks of delay-line storage, making a total
of 1152 blocks.

There are two addressing systems used in the main store. In the simpler system the locations are
numbered straight through from to 8191; these addresses will be called decimal addresses. In the
other system, which is more generally useful, each location is specified by giving its block'number
and its position within the block. The blocks are numbered to 1023. Within each block the eight
individual locations are given position-numbers between and 7. Thus we may refer to a certain
location as being in block 342 in position 5; this would be written 342.5, which is the block-and-
positlon form of the address (this word would also have a decimal address 2741). The posit ion- number
may be thought of as an octal digit.

The block-numbers are sometimes referred to as block-addresses; this is an extension of the term
"address" since it is here used to identify blocks rather than individual locations. The block- addresses
are often prefixed by the letter B, thus we may refer to block 342 as B342.

It has been arranged that sixteen of the tracks are isolated, i.e. the corresponding heads may be
used for reading but not writing. These 16 tracks form two Isolated stores, each of 128 blocks (1024
locations) and both addressed from 896.0 to 1023.7 (or 7168 to 8191 on the decimal address system)
either store may be selected by means of one of the engineers' switches on the control desk under the
hinged flap. These isolated blocks are used to store permanently certain useful programmes; the
Initial Orders, which are of great value to the programmer, are in one of the stores, and in the other
are the engineers' test programmes. When the appropriate switch is in the normal position, the
isolated store containing the Initial Orders is selected. The rest of the main store consisting of
896 blocks (or 7168 locations) addressed from 0.0 to 895.7 (or to 7167 on the decimal address system),
is available for other programmes. tt

It should be noted that information stored on the drum, but not in the delay-lines, is not
"volatile", i.e. it remains there even when the computer is switched off at the mains; in particular
the programmes in the isolated part of the main store are always there and can be considered almost a
part of the machine.

A magnetic drum combines economy with a reasonably large storage capacity; this is why it is
used in many computers. It suffers however, from the disadvantage of a relatively long access-time,
which is the time needed for information to become available. One will usually have to wait until
some particular part of the drum is under the heads before one can actually read or write. This time
will never exceed one revolution time (16 rallllsec in Pegasus) and this is therefore called the
maximum access-time; the average waiting time will be about half a revolution (8 millisec in Pegasus)
and is called the mean access-time, if we neglect the time occupied by the actual reading or writing.
On some machines the head selection is done by relatively slow relays and time must be allowed for
these to operate when changing from one head to another; on Pegasus this switching operation is done
electronically and no time need be allowed for it.

If the speed of a magnetic drum computer is not to be severely limited by the access-time of the
drum it is essential to provide as well some other kind of storage having a shorter access-time. This
is sometimes called the "fast" store and various other terms are used, such as "working" store or "quick
access" store or "high-speed" store; but it is referred to as the computing store in Pegasus, for
reasons which will become apparent later.

As a consequence of investigations into the amount of waiting time spent in some of the standard
programmes run on Pegasus 1 (i.e. the time spent in waiting for the drum to be in the correct position
for a word to be read or written), the first 16 blocks of main store in Pegasus 2 are held on 8-word
delay-lines.* This has the effect that the transfer of words between the computing store and this
part of the main store is carried out almost immediately without having to wait for the drum to come

t Pegasus 1 has a drum storage capacity of 9216 words, but the earlier model of Pegasus 1 has a

smaller drum which has a storage capacity of 5120 words. In what follows we refer specifically to
Pegasus 2.

tt On the small drum version of Pegasus 1, there is one isolated store of 128 blocks: the non-isolated
part of the store has a capacity of 512 blocks (4096 words) and is referred to as the M96-word
store. (The main store on the present Pegasus 1 and Pegasus 2 is referred to as the 7168-word store).
A version of the Initial Orders, which lacks some of the facilities of the 7168-word store Initial
Orders, Is stored in the first part of the isolated store from B512, and the engineers' test
programmes are stored beyond this.

* A further 16 blocks of main store held on 8-word delay lines may be provided in Pegasus 2 as an optional extra.

- 14 -

PEGASUS

2.3

into position. These 16 blocks (BO to B15) have exactly the same addressing system as they would have
if they were held on the drum, so that to the programmer the only difference is the increase of speed.
(Further details of the time are given in section 3.10).

2.3 The computing store

The computing store of Pegasus is made up of registers, each of which can store one word (its
content)', the registers correspond to the storage locations in the main store.

Most of the registers are -made up of circulating magnetostrictive delay-lines and amplifiers. Each
delay- line consists of a length of nickel wire with a coil near each end. When a pulse is passed
through one of these coils the nickel inside it shrinks momentarily (magnetostrictive effect) and
compression waves (i.e. sound waves) travel along the nickel wire in both directions away from the
coll. One of these waves is absorbed and the other one travels down the nickel wire to the second
coil, in which it Induces a small pulse. This small pulse can be amplified and, after having any
distortions removed, can be fed back to the first coil; the whole process is then repeated. In this
way a pulse can be kept circulating indefinitely, provided the power is switched on to the amplifier.
The circulation- time is one word-time, i.e. X26 microsec, so that a whole word of pulses can be kept in
circulation; any gaps in the train of pulses will of course persist. At any particular instant the
39 pulses (and "no-pulses") making up a word (together with the three gap pulses) will be strung out
along the nickel wire, travelling down it with constant velocity.

The word stored in the delay-line is available at the output of the amplifier; it may be read as
many times as desired without disturbing it. In order to replace it by another word we have only to
break the circulation loop for exactly one word-time while the new word is being fed into the delay-
line. This is shown diagrammatic ally in Pig. 2.1.

Delay line

Amplifier

■4

Read out

Write in

Pig. 2.1 A delay-line storage unit

Many computers incorporate del ay- line stores of one form or another. In some of these each delay-
line holds more than one word (as in the 8-word delay-lines comprising the first 16. blocks of main store
in Pegasus 2); this increases the storage capacity at the expense of increasing the access- time. With
single-word delay- lines, such as are used in the computing store in Pegasus, each stored word can be
regarded as available at any time.

The registers making up the computing store are divided into three groups:

(a) the ordinary registers,

(b) the accumulators,

The ordinary registers make up the bulk of the computing store; there are 48 of them, arranged in
six blocks, each of eight registers. The addresses of the ordinary registers are written in the same
way as the addresses of locations in the main store, e.g. the register in position 3 of block 4 has the
address 4.3. If there is any danger of confusion we shall prefix computing store addresses by the letter
U and main store addresses by the letter B; thus t/5.2 is the address of an ordinary register, and B5. 2
that of a storage location in the main store. The blocks in the computing store are numbered UO to U5
so that the first of the ordinary registers is UO.O and the last is f/5.7.

The eight registers called accumulators bear the addresses to 7; these addresses are often
prefixed by the letter X (e.g. X2). The accumulators have special properties which single them out from
the other registers; these will be described later. Accumulator is sometimes called the dummy
accumulator, its content is always zero and cannot be changed; this accumulator is not made up of a
delay-line. Each of the seven other accumulators can be used to store any word required.

The special registers have the addresses 15, 16, 17, 24, 32, 33, 34, 35, 36 and 37. t They are not
made up of delay- lines and are used for special purposes which will be described later (see Section
2.9).

We shall often need to refer to the content of a register, i.e. to the word held there. We shall
write C(5.2) for the content of ordinary register 5.2, C(6) (or sometimes xg) for the content of accumula-
tor 6, and so on.

There are facilities for transferring words from the main store to the computing store and vice-
versa. These words may be transferred either singly or in blocks of eight words.

Pig. 2.2 (page 16) shows the two stores of the basic Pegasus computer; some of the details will be
explained later. In this figure the address of each of the registers of the computing store is shown;
these addresses are always written in the way indicated; for example 3 means accumulator 3, and 3.0
means ordinary register 3.0. The reader should endeavour to keep this picture of the computing store
in his mind while he reads the next few chapters; he should also note particularly that the word
register applies not only to the ordinary registers but also to the accumulators (as well as the special
registers).

t In the small drum Pegasus 1, registers 24 and 37 are not present, and register 36 only on those
installations with magnetic tape. Both registers 36 and 37, but not 24, are included in the large
drum Pegasus 1.

- 15

2.3

PEGASUS

COMPUTING STORE

Name Address

of register of register

Accumulators

(in)

Special
Registers

(71

J/4

32
33
34
35
36
37

0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7

(75

Always zero
J single-word
I transfers

P C9
1-1 iS

From

input

devices

To
output
devices

Y Y 1 A '

Double-length

Handswitches (20 digits)
Input/Output (checked) —

3: Input/Output (unchecked)

Checking
equipment

External Conditioning Setting in 24£

1.
1.
1.
1.
1.
1.

i.
1.

2.
2.
2.
2.
2.
2.
2.
2.

3.
3.
3.
3.
3.
3.
3,
3.

4.
4,
4.
4,
4.
4,
4.
4.

5.6

5.7

Always -1.0
Always 1/2
Always 2"^°
Always 2"^^
Always 2"^*
Always 7/8

OVR □ Overflow Indicator

MAIN STORE

-Block and
position-)

■P Q
3 <

E B

6893

6894

6895

% w

B 896

B1023

Decimal
address

5
. 6
. 7

9
, 10
11
.12
. 13
. 14
. 15
. 16
. 17
.18
.19

7150
1
2
3
4

7155
6
7
8
9

7160
1
2
3
4

7165
6

7167

896 blocks
of 8

words each
=7168 words

2 sets of

isolated

blocks each

containing

128 blocks

of 8 words

each (1024

words)

= 2048 words

Fig. 2. 2 The Computing Store and Main Store of Pegasus

PEGASUS 2.4

2.4 Outline of operation

The orders or instructions which the computer obeys in the course of carrying out a programme are
represented in coded form by words; such words are referred to as order-pairs in Pegasus, since they
each represent two orders.

The control unit of the computer, which controls the whole machine, selects the orders to be
obeyed exclusively from the ordinary registers in the computing store. The address of the ordinary
register concerned is called the order- number and is held in a special order-number register (or O.N.R. )
in the control unit. There is also an order-register (or O.R. ) to hold the orders currently being
obeyed.

The sequence of operations while the computer is obeying a programme is as follows:

(a) The control unit determines the order-number (from the O.N.R.) and connects the ordinary

register specified to the order- register (O.R. ). This causes a copy of the specified
order-pair to flow into the order- register. This operation requires one word-time.

(b) The first order (a-order) of the order-pair is obeyed, i.e. the computer carries out an

elementary operation determined by the digits of the order. This operation usually takes
two word-times, but some orders (e.g. multiplication) require more than this.

rule. While this order is being obeyed the order-number in the O.N.R. is increased by
unity, so that the next order-pair will be selected from the next ordinary register.

This cycle of operations is normally i-epeated a number of times. It will be seen that it usually
occupies 5 word-times, during which two orders are obeyed; we can say therefore that the average time
of obeying one (simple) order is 2% word-times, or 0.315 millisec. Alternatively we can think of the
a-order as occupying 3 word-times and the 6-order 2 word-times; this is often a more useful approach,
and it is the one we shall adopt. The orders obeyed are selected from consecutive registers and are
therefore said to be obeyed sequentially.

Certain orders, called jump orders, may interrupt the regular sequential selection of orders and
cause the machine to start obeying orders from some specified register. They do this by changing the
order-number in the O.N.R. .and there are arrangements for jumping to or from either order of a pair.

It is important to realize that the computer selects its orders exclusively from the ordinary
registers in the computing store; there are no arrangements for taking orders directly from the
accumulators or from the main store. The usual process is to place the programme in the main store
and then to transfer a few blocks of it into the computing store to be obeyed; when these orders have
all been used up a further instalment is brought in from the main store, and so on.

Apart from a few orders effecting transfers to and from the main store, all the orders are
concerned only with the computing store. All the arithmetical operations and organisational work are
carried out in the computing store, and it is here that sections of the programme and numbers currently
required are stored. Since there are no problems of access-time in the computing store it follows that
the speed of operation of the machine is high. The computer is thus organised on the basis of a tvo-
level store; all the work is done in the "working space" of the computing store, the main store being
used largely to hold orders and numbers not immediately required.

In this way a high speed of operation is obtained despite the use of a magnetic drum and without
resorting to optimum coding (this Is an alternative system in which orders and numbers are placed in a
long delay-line or round a drum in such a way as to be available when required). If the pTogrammer has
to use optimum coding then he must consider timing matters while he is writing the programme. Some
computers are provided with a single-level store, i.e. there is only one store; this is much simpler
from the programmer's point of view, but in most such computers the size of the store is limited because
of its high cost (if it has Immediate access) or there may be the necessity for optimum coding. In
larger computers than Pegasus different considerations apply, of course.

2.5 The written form of an order

A single Pegasus order is made up of four parts; a typical one is written as follows:

3.1 2 01 4

In this order 3.1 is the N-address (or first address); it is here the address of one of the ordinary
registers in the computing store. The second part of the order is called the X- address (or second
address); in this order it is 2, indicating that the accumulator X2 is concerned. The function of
this order is 01, which specifies a certain addition operation. The last part of the order is 4,
which shows that the action of the order is to be modified by the content of X4 before being obeyed;
this part of the order may be called the modifier- address (or M-part).

We shall confine ourselves at present to unmodified orders, i.e. those whose modifier- address
is zero. In such orders the content of XO is used for modification and no change occurs (it will be
remembered that XO is the dummy accumulator, whose content is always zero). If the modifier-address
is zero there is no need to write it in; this part of the order may be left blank. For the moment
therefore we shall write down only orders containing three parts.

Let us consider the effect of the order

3.1 2 01

when it is obeyed. This order causes the computer to take the word in the ordinary register 3. 1 and
to add it to the word in accumulator 2, leaving the result in X2. In other words, the content of X2
after the order has been obeyed is the sum of the previous content of X2 and the content of 3. 1. This
process does not disturb the content of 3.1 which will be the same before and after the operation of
the order. After obeying the order in this way the computer will proceed to the next order. The
order written

0.7 6 01
will similarly add C(0. 7) and C(6) and leave the sum in XG, the expression C( ) denoting the content.

2.6 PEGASUS

Both these orders have the function 01; in such orders the J-address (the second part of the order) is
always the address of one of the accumulators. The yv- address, on the other hand, may be the address of
any register in the computing store (i.e. it may refer to an ordinary register, an accumulator, or a
special register). For example, the order written

3 5 01

causes the computer to add C(3) and C(5) and leave the result in X5, and the order

7 7 01

will add C(7) to itself, i.e. it will double C(7).

We shall now introduce a notation which is useful for writing down the effects of various orders.
We shall write N and X for the A^- address and J- address respectively in an order. Thus in the order

3. 1 2 01

N is 3. 1 and X is 2, or we may write N = 3.1 and X = 2. In general N and X represent the addresses of
whichever register and accumulator are written in the order. The content of A' will be written as n, and
the content of X as x, so that in the above order n = C(3. 1) and x = C(2)t. These represent of course
the contents before the action of the order; we shall use n' and x^ to represent the contents after the
order has been obeyed. With the aid of this notation we can write the effect of an order with the
function 01 as follows:

01 x' - X + n.

This equation simply states that the content of the accumulator specified by the X-address after the
order has been obeyed is equal to the content of this accumulator before the operation of the order plus
the content of the register specified by the ;V-address. Since n' is not referred to it is implied that
the content of the register specified by the A^-address is unaltered by the order; except of course for
the special case in which N = X, as in the order

7 7 01

It is of course also understood that all the remaining registers are undisturbed.

The words on which an order operates are called the operands. In an order with function 01 the
operands are the contents of the register and accumulator specified in the order. When describing the
effects of various orders it is often convenient to refer to them by their function parts; thus we
shall talk of an 01- order when we mean an order whose function part is 01. The function part of an order
is sometimes denoted by F.

It should be noted that the equation defining the effect of an 01-order is not affected by the
conventional position of the binary point in the word, provided of course that we put it in the same
position in all the words occurring. In particular, the words may be interpreted as numbers in either
the fractional or the integer convention.

All the orders so far described have been Ol-orders. In general the function of an order is
written as a pair of octal digits, i.e. digits having values between and 7. We may therefore get
orders whose functions are 00, 01, 24, 53, 10, 65, 77, and so on. The operations caused by these
different orders will be described in this and the next chapter, whose subject is the order- code of
Pegasus, i.e. the catalogue or list of available functions and their effects.

The various operations which can be carried out by obeying an order fall naturally into groups.
The group to which an order belongs is identified by its first function digit; thus the orders of group

6 are those whose function parts are written 60, 61, 62 67. The 01-order belongs to group 0. The

effects of the orders in various groups may be roughly summarised as follows;

group Simple copying or arithmetical operations, the result being left in an

accumulator,

group 1 Similar to group but the result is left in a register,

group 2 Multiplication and division.

group 3 Unassigned (i.e. a spare group) except for 37, see chapter 11.

group 4 Simple operations with integers.

group 5 Shifts (i.e. multiplication and division by powers of 2).

group 6 Jumps,

group 7 Transfers between the main and computing stores.

Some of the functions are not used, e. g. 07, 30, 75; orders with these functions are referred to
as unassigned orders. The computer stops if it encounters such an order, and illuminates a special
light (marked "unassigned order") on the control panel.

The whole order-code is summarized, for reference, in Appendix 1 and on a single sheet at the back
of the book, and it is recommended that a beginner should refer to these frequently when starting to
write programmes; it is not necessary to make the effort of learning the whole order-code by heart.
The systematic arrangement of the orders make them easy to remember.

Before discussing any further how orders are written and what they do we shall describe briefly
how orders are represented inside the computer.

2.6 The Internal form of an order

Inside the computer an order is represented by 19 binary digits; these we shall number 1 to 19,
counting as usual from the left (i.e. from the most-significant or ms digit). The 7 bits numbered 1 to

t Occasionally it is useful to write X2 Instead of C(2), etc.

PEGASUS

2.6

7 are used to represent the yv-address in the order; they may be called the Af-digits of the order.
The next 3 bits (digits 8,9,10) represent the A^-address directly in binary; thus it X = 5 these three
bits are 101. The function of the order is represented by the six f-digits (11 to 16) three of which
are used for each octal digit in the written form of the order; this is illustrated by the following
table: -

Written function

01
24
53
67

f-digits

000 001
010 100
101 Oil
110 111

The last 3 bits of an order (digits 17, 18. 19) represent the modifier-address directly in binary; they
are all zero in an unmodified order; these digits are sometimes called the ^-digits in the order. The
way in which the 19 bits of an order are allocated may be shown diagrammatically as follows.

We shall describe later, in detail, the way in which the yv-digits are used to represent the A^-address
(see Section 3.12); at present we need only state that if the Af-address is one of the accumulators
then the A^-digits give the address directly in binary. For example, the order written 2 5 01 is
represented in the computer by the following 19 binary digits:

0000010 101 000001 000

N=2 X=5 F=01 Af=0

Each of the two orders in a 39-bit order-pair occupies 19 bits. The left-hand or ms binary digit
of the word does not belong to either of the orders; this digit (number 0) corresponds to the sign-
digit of a number-word and is called the stop/ go digit, its use will be described in Section 3.9. The
a-order of the order-pair occupies digits 1 to 19 of the word, as described above; the 6-order
similarly occupies digits 20 to 38, these being used in the same way as corresponding digits of the
o-order. The way in which the digits of an order-pair are allocated may be shown diagrammatically as
follows.

1 2

stop/g
digit

a-order

6-order

It will be recalled that a simple order (such as an 01-order) is obeyed in two word-times. We
shall now briefly describe the sequence of operations which occur when the order

3.1 2 01

is obeyed. The order will at this point be held in the order-register, and here its function digits
(000 001) are decoded and used to set in motion a train of events. The first thing which occurs is that
the mill is set up to do an addition; simultaneously the registers containing the two operands (i.e.
3.1 and X2) are connected to the input of the mill; these operands enter the mill and are added during
the first word-time. During the second word-time the result of the operation comes out of the mill and
is sent to its proper destination, viz. X2, where it replaces the previous content.

It is most important to distinguish the written form of a programme from its representation
inside the computer. The written orders of a programme consist of ink or pencil marks on paper, and
they are expressed in a way which has been chosen for the convenience of the programmer. Inside the
machine the programme is represented in the form of trains of pulses or as magnetised areas on the drum.
A conversion process has to be applied to the written orders before they can be stored or obeyed by the
computer. The first step in this process is the "typing" out of the programme on a teleprinter (or a
keyboard perforator), which provides a length of punched paper tape and a printed sheet (for checking
purposes). The paper tape can be "read" by placing it in a tape- reader attached to the computer and
calling in the Initial Orders. These are a permanently available programme stored in the isolated part
of the main store; the computer can be caused to obey this programme by operating a special key on the
control panel; this key is called the Start key, and the way in which it is used will be described in
Section 4.3. The Initial Orders cause the computer to read the tape, one character at a time, and to
build up the orders, which it places in the main store in the form required by the machine. The way in
which the punched form of the orders is converted to the stored form depends entirely on the Initial
Orders, and has been chosen so as to make the programme as convenient as possible to write and to punch.
The main function of the Initial Orders is to read, convert and store programmes; it is used every
time a programme is put into the computer. When the whole programme has been stored in the main store
the Initial Orders can be caused to transfer a part of it to the computing store and to start the com-
puter obeying it. This subject is discussed further in Section 4.3.

19 -

Effect

X + n

—n

X - n

n - X

2.7 PEGASUS

2. 7 The orders of groups and 1

The orders of groups and 1 are concerned with simple operations such as copying, addition and
subtraction. The Ol-order has already been described; it will be recalled that its effects may be
briefly summarized by the eauation x' = x + n. The 00-order is even more simple; it is described by
the equation

00 «' = n,

which means that the content of the specified accumulator is replaced by a copy of the content of the
specified register. For example, the order

5.4 3 00

causes the word in ordinary register 5.4 to be copied into J3, the previous content of ^3 being lost.
The word in 5.4 is not changed by this order.

For example, if we have two numbers in ordinary registers 5.0 and 5.1, we can add them and leave
the sum in X5 by the following two orders:

5.0 5 00 first number to ^5

5. 1 5 01 add second number

The result of any order of group is always left in the specified accumulator. The first five
orders of this group may be summarized as follows. Here the word "register" indicates the register
(i.e. ordinary or special register, or accumulator) specified by the iV-address written in the order.

Description

Replace content of accumulator by copy of content
of register.

Add content of register into accumulator.

Replace content of accumulator by minus content
of register.

Subtract content of register from accumulator.

Subtract content of accumulator from content of
register, leaving the difference in the accumulator.

For example, suppose we have three numbers a, b and c in the ordinary registers 4.0, 4.1 and 4.2
respectively. We can form the quantity a + 6 - c in J2 by the orders

4.0 2 00 a to X2

4. 1 2 01 add 6, result is a + 6 in X2

4.2 2 03 subtract c, final result in 1^2.

Alternatively the following orders could be used:

4.2 2 02 -c to X2

4.0 2 01 add a, result a - c

4. 1 2 01 add b. result a + b ~ c

It is clear that there are many equally good ways of doing this operation.

The orders of group 1 are similar to those of group but the result of the operation is always
left in the specified register. , The most important order of group 1 is the 10-order (read as "one-oh",
not as ten):

10 n' = X,

which means that the word in the register is replaced by a copy of the word in the accumulator. For
example, the order

2.4 7 10

replaces C(2.4) by a copy of C(7). The content of the accumulator is unchanged by this order.

As an example, suppose the numbers a, 6 and c are, as before, in 4,0, 4.1 and 4.2; we can place
the quantity a - b - c in 5.0 by the orders;

4.0 5 00 a to X5

4. 1 5 03 subtract 6, result a - 6 in X5

4.2 5 03 subtract c, result a - 6 - c in ^5
5.0 5 10 a - 6 - c to 5.0

In this Illustration we have used ^5 to form the result; we have to use an accumulator for this, but
any accumulator would have served equally well.

It will be seen that the 10-order is similar to the OO-order except that the roles of register and
accumulator are interchanged. This analogy holds good for the remaining orders of group 1, The effect
of any order of this group can be derived from the corresponding order of group by lntercha,nging the

20 -

PEGASUS

2.7

words "accumulator" and "register" in the verbal description, or by interchanging the letters n and x
in the defining equation. For example, the 01-order is defined by the equation

01 a:' = x + n,

and the 11-order (read as "one-one") by the equation

= n + *.

In this order the number in the accumulator is added into the register. Thus, the order

0.6 3 11

causes the sum C(0.6) + C(3) to appear in 0.6; the content of X3 is unchanged. It will be seen that
the ordinary registers may be used for addition and subtaction. The first five orders of group 1 may
be summarized as follows.

Description

Replace content of register by copy of content of
accumulator.

Add content of accumulator into register.

Replace content of register by minus content of
accumulator.

Subtract content of accumulator from register.

Subtract content of register from content of
accumulator, leaving the difference in the register.

As an illustration, suppose we have two numbers in 4.0 and 4.1, and we have to replace the second
number by their sum. We must use an accumulator for intermediate storage, let us use J4, and the
following orders will do:

4.0 4 00 first number to J4

Eff

ect

n + X

-X

n - X

X - n

4. 1 4 11

sum to 4. 1

As a further example, suppose a, b and c are the numbers in 4.0, 4.1 and 4,2; the following orders
will replace 6 by a + 6 in 4. 1, and c by c - a - 6 in 4. 2:

4.0 3 00

4.1 3 01
4. 1 3 10

4.2 3 13

a to X3
a + b to X3
a + fc to 4. 1
c - (a + 6)

a - b to 4.2

It sometimes happens that we wish to clear a register, i.e. to replace its content by zero. We
use the dummy accumulator JO for this. To clear an accumulator, say J4, we can use the order

4 00

and to clear an ordinary register, say 2.0, we use the order

2.0 10

There are, in nearly all these examples, many other solutions which are equally "good", i.e. which
require as few orders. If possible we always prefer the most direct and simple method. Thus to form
in J4 the difference o - 6 of a (in 4.0) and b (in 4.1) we would prefer the orders

to the orders

4.0 4 00
4. 1 4 03

4.1 4 00
4.0 4 04

a to J4

a - b to X4:

b to J4

a - b to X4.

If the numbers we are dealing with are in accumulators we prefer orders of group to those of group 1;
thus the orders

3 2 10

6 5 13

are equivalent to the orders

2 3 00
5 6 03

but the latter would be preferred. The choice between such alternative orders is largely an aesthetic
matter, but if unconventional methods are used the checking of programmes is made much more difficult.

2.8 PEGASUS

In the equations, such as

x' = X + n,

which define the effects of the orders of groups and 1, the letters x and n represent the contents of
the registers used; these may be Interpreted according to either the fractional or the Integer
convention, provided of course that only one convention is used throughout any particular equation. In
the orders 00 and 10 the operands need not have any particular significance - they may be fractions or
integers or words to which no numerical value is attached (e.g. order-pairs).

Note that it takes two orders to add together two numbers in ordinary registers but only one
order to add two numbers in accumulators. This is just one of the differences between the accumulators
and the other registers.

The remaining orders of groups and 1 are described in Section 3.11.
2.8 Writing the programme

The orders of a programme are written on printed programme sheets in the columns provided; an
order-pair occupies two lines on the sheet. When the programme is eventually obeyed the order-pairs
occupy ordinary registers in the computing store, and we usually write to the left of each order-pair
the address of the register which will hold it at that time. For example, two order-pairs obeyed from
0.0 and 0.1 in the computing store, might be written as follows:-

5.0

1 00

0,0

5.1

1 01

5.2

1 03

0.1

3.6

1 10

Since the computing store is seldom large enough to hold the entire programme and the numbers on
which it is to operate, these are all placed in the main store during the input process. When the
whole programme (and, probably, the numbers) have been read in and placed in the main store, it is
arranged that the first four blocks of the programme are copied into the computing store. This is all
done by the Initial Orders programme, which also arranges for the computer to start obeying the orders
at a specified point (this is called entering the programme). Consequently the programme has to be
divided up into blocks, each of 8 order-pairs (i.e. 16 orders), and the programmer has to Insert block-
transfer orders in the programme at suitable places so as to read fresh blocks of orders from the main
store into the computing store when those already in the computing store are no longer needed. Two or
three of the six blocks of ordinary registers in the computing store are usually reserved for those
parts of the programme which are currently being obeyed; the remaining blocks are used for constants,
data, intermediate results, etc.

When starting to write a programme one cannot usually forsee exactly how best to divide it up into
blocks. Parts of the programme are therefore sketched out in pencil on squared paper before being laid
out in blocks on one of the printed programme sheets.

Figure 4.3 on page 72 shows a four-block programme sheet with the orders written in. It will be
noticed that the addresses of the registers holding the order-pairs are abbreviated; the address is
written in full against the first order-pair only in each block. The "box" above each block is used to
Indicate the main store block-number where the block of orders is held. Many of the details of this
programme will be explained in later chapters. Programme sheets in general use are laid out with 2, 4
or 6 blocks.

Sometimes the orders needed will not completely fill a block. In this case unused words may be
filled with zeros, or extra "dummy" orders may be inserted. A dummy order is one which has no effect,
e. g. the order

which simply copies C(0) into XO. This order is the one conventionally used as a dummy order; it may
be written on a programme sheet simply as a zero in the ^-address column.

Numbers may be written on the programme sheets as well as orders. They may be either fractions or
integers and each number will of course occupy the whole of a register in the computing store (i.e. two
lines on the programme sheet). A number must be preceded by its sign (+ or -) since this is used to
distinguish it from orders. If it is a fraction the decimal point must be written. Numbers and order-
pairs may be intermingled on the programme sheets, but the programmer must take care that the computer
does not try to "obey" a number; unconditional jumps may be used to avoid this. As an illustration of
how numbers may be written, let us suppose that the fractions 1/4 and -1/8. and the integer 1000 are to be
placed in X2, 3 and 4 respectively at some stage in a calculation. The relevant part of the programme
sheet might appear as shown on the facing page, where we assume that the computer starts to obey the
orders in 0.3 (as indicated by the arrow).

Numbers written as a part of a programme in this way are often called constants, since their values
are not altered. Usually only occasional constants that are required at some stage in a calculation are
written on programme sheets. If the problem uses many numbers they would not be written on the programme
sheets but all together on other sheets of paper.

The numbers and order-pairs are punched more or less as written on the prograjnme sheets, and the
resulting tape is read by the Initial Orders programme, which converts the items on the tape into the
appropriate binary form and places them in the main store. We shall describe later the details of the
punching; at present all we need say is that every number must be preceded by its sign and must be
either a fraction (with a decimal point) or an integer. A fraction may have up to 11 decimal digits after
the point, and will be converted to the nearest binary equivalent (with an error of at most % e, see

- 22

PEGASUS

2.9

Section 1.8). An integer is converted into binary without error, as also is a fraction which can be
precisely represented by a 39-bit word (e.g. 0.25 or -0.625).

B 2 1

0.0

+0.25

-0. 125

+1000

oTo

00^

0.1

0.2

= 1/4

. fractions

= -1/8,

integer

Place 1/4 in X2

Place -1/8 in J3

Place

1000 in X4:

We often rule a vertical line to the left of a number written on a programme sheet:
to emphasise that it is not an order.

this is done

2.9 The special registers

The special registers are a group of registers in the computing store which are used for special
purposes, they are not made up of delay- lines and cannot be used for storing words in the same way as
the accumulators or the ordinary registers. They have the addresses 15, 16, 17, 24, 32, 33, 34, 35, 36,
37 (some other addresses are used vrtien certain extra equipment is fitted to the basic computer). These
registers are shown in the diagram of the store of the computer (Pig. 2.2).

Registers 32 to 37 hold permanently certain useful constants. For example, register 33 holds 'A so
that the order

33 4 01

has the effect of adding ^ to the number in J4. The order

33 4 10

has no effect. The contents of these constant registers are as follows:

-1.0

1/2

2-10

2-13

2-16

7/8

It would of course be easy to place one of these constants in a register by writing it on the programme
sheet along with the orders which need it. These particular numbers are needed fairly often however,
as will appear later; it should be noted that each of them (except that in 37) is represented by a word
only one of whose 39 digits is a one.

The content of register 15 is determined by the settings of a row of switches on the control panel;
these are referred to as the handswitches. Twenty of these switches are referred to as HO, HI, H2, ....
H19 and each of them may be either up or down. The up-position of each switch corresponds to and the
down-position to 1, so that we may set up any twenty-digit binary number by pressing down the switches
where ones appear in the number. When this has been done an order such as

15 4 00

will place in X4 a word whose first 20 digits are determined by the handswitches; the remaining 19
bits of this word are all zero. For example, if ffl9 only is down the above order will place 2"^' in ^4.
We write H19 = 1 to indicate that the switch H19 is down. This kind of order is said to read the
handswitches. An order such as

15 4 10

has no effect. The handswitches can be very useful to the programmer for manually directing the
operations of the programme; how they are used will be explained later.

Registers 16 and 17 are used for input and output and will be described later. Register 24 stores
the current setting of the external conditioning relays (described at the end of Section 3.10).
Registers 20 and 21 are used only on those computers equipped with magnetic tape auxiliary storage.
The content of any other register, such as 31. is always zero; such registers have the unallocated
addresses between 8 and 63. The addresses 64 onwards are actually those of the ordinary registers; it
is much more convenient to use the customary notation for these, however.

- 23 -

2.10 PEGASUS

2. 10 The orders of group 4

By using the Orders of groups and 1 together with numbers placed in ordinary registers along with
the programme we can carry out many useful operations. In fact most of the orders in the majority of
programmes are concerned with simple operations such as addition, subtraction and copying. The special
constant registers relieve us of the need to punch and store certain useful numbers.

Most programmes require small integers for a variety of purposes. Such integers could of course be
punched on the programme tape and read into the computer along with the programme; but this would be
wasteful of storage, especially in the computing store, since many small integers are required during the
course of a programme. This procedure would also be inconvenient. The orders of group 4 can be used to
produce any small integers as they are needed. These orders closely resemble the orders of group 0; but
the iV-address (or first address) in the order is not any longer an address, but is actually the integer
required. For example, the order

(27) 3 40

causes the integer 27 to be placed in J3. The number written on the left in the order is usually
encircled so as to emphasize the fact that it is not the address of any register. This number can have
any value from to 127 (since it is represented by 7 binary digits inside the computer).
The effect of the 40-order may be written

x' = N,

where N stands for the encircled number written first in the order. Note that A^ is to be carefully
distinguished from n, which is the content of a register. In this equation the words are, of course, to
be interpreted on the integer convention.
The order

4 6 00
copies C(4) into XG, and the order

6 40

places the integer 4 in X6. The circle round the number is merely an aid when reading programme- sheets,
no special punching is used to indicate that the number is encircled. In fact the above two orders are
differentiated only by their function digits when they are punched on tape or stored inside the computer.

It will be seen that the 40-order is similar to the 00-order in that the result is left in the
specified accumulator. The first operand is, however, explicitly written in the order and is not the
content of any specified register. This analogy also holds for the remaining orders of group 4. The
effect of any order of this group can be derived from the corresponding order of group by replacing
the words "content of register" by "number written in the order" in the verbal description. This is
equivalent to replacing n by A^ in the defining equation. For example, the 01-order is defined by the
equation

01 x' = X + n,

and the 41-order by the equation

41 x^ = X + N,

where it is understood that the numbers concerned are integers.

Where necessary we shall put a suffix j or ^p on letters such as n or x to denote that the
corresponding words are to be interpreted as integers or fractions respectively. For example Xf means
the fraction x, and xj means the integer x. The above equation could be written:

x/ = xj + N.

The first five orders of group 4 may be summarised as follows.

Description

Replace content of accumulator by integer written
in the order.

Add integer written in the order to integer
in accumulator.

Replace content of accumulator by minus the
integer written in the order.

Subtract integer written in the order from
integer in accumulator.

Subtract integer in accumulator from integer
written in order; result in accumulator.

In this summary the equations are all written on the integer convention; if we use the fractional
convention they must be written differently, for example the equation of the 41-order becomes

Effect

■';

Xj + N

-N

«;

Xj - N

N - Xj

41 x^ = xp. + iV.2-38,
Xf. = xp + N€.

- 24

PEGASUS 2.11

As an example suppose we wish to use the integer 49 for counting purposes, it can be set initially
into accumulator 2 by the order

@ 2 40

To subtract 1 from the integer in X2 we would write

02 43
The remaining orders of group 4 are described in Section 3. 11.

2. 11 Overflow

According to the standard fractional convention any number x represented by a word must be between
-1 and +1, in fact such a number must satisfy the inequalities

-1 ^ a: < 1.

It may happen during arithmetical operations that numbers are produced which exceed capacity, i.e. lie
outside the permitted range of values. This is called overflow. For example, overflow will occur if we
add 0.71 to 0.95. In such an event the computer will produce a wrong result and a warning is given by
a special overflow- indicator. This overflow- indicator is usually referred to as OVR; it is a two-state
device which is normally dear but will be set when overflow occurs. Once OVR has been set it will
remain set, regardless of other operations, until certain special orders are obeyed (see Section 3.8).
One of the lights on the control panel shows the state of the overflow- indicator.

A "Stop on Overflow" key is provided which, when depressed, will cause the computer to stop on the
completion of an order during which OVR has been set. On switching to STOP and RUN the computer will
continue working as though the stop had not occurred, and OVR must be cleared and set again before the
stop can occur again t.

When OVR is set it is impossible to transfer words to the main store; the computer will stop if
it comes to an order calling for such an operation. This means that the numbers obtained from a sequence
of calculations cannot be written away into the main store if overflow has occurred at any stage. Since,
as a rule, the programmer will not want numbers to overflow this stop prevents the wastage of machine
time which would occur if much further calculation were done with the wrong numbers. A special light
on the control panel indicates this kind of stop; it is marked "writing with overflow".

If overflow occurs during any order except a division order, then the result actually obtained will
differ from the correct result by 2 or a multiple of 2 (on the fractional convention). For example, if
we attempt to add 0.71 to 0.95 we shall get -0.34, which is 2 less than the correct sum 1.66, and OVR
will be set. Overflow can, of course, also occur with integers; in this case the result actually
obtained will differ from the correct result by a multiple of 2^^.

Overflow can be thought of as a loss of significant digits at the left (or ms) end of the word;
this is usually a serious matter unless the programmer has foreseen it, which may be difficult in a
complicated programme. Overflow is usually caused by insufficient scaling-down of the numbers occurring
in the calculation. One cannot lay down any definite rules about the action to be taken if overflow
occurs, but the overflow- indicator can be sensed by the programme and it may be possible to arrange that
the offending numbers are all automatically scaled down and the appropriate part of the calculation
repeated.

Apart from orders causing writing into the main store (functions 71 and 73), the only orders
affected by the setting of OVR are those with functions 23 (justify) (see Section 3.4), 64 and 65 (see
Section 3.8). The results obtained from any other order will be the same whether OVR is set or clear.
Note that the orders 00, 10, 40 and 42 cannot set OVR; the orders 02 and 12 will set OVR only if the
number concerned is -1.0 (on the fractional convention).
)f When the computer obeys an order such as

33 5 11

the sum, i.e. C(33) + C(5) = % + C(5), is actually formed in the mill, although C(33) is, of course,
unaffected. An order of this kind may, however, set OVR. Similar remarks apply to the other constant
registers and to register 15 (handswitches) . This can sometimes be turned to good account; the above
order will, for example, set OVR only if C(5) > % and can be used to indicate this fact without
disturbing the contents of any registers. The order

32 02
4 is occasionally useful; it sets OVR but has no other effect.

In the case where overflow is caused by a 20- or a 21-order (i.e. -1.0 x -1.0) which is obeyed in an
a-order, the stop will occur after the fc-order, i.e. the computer will be waiting to obey the next
a-order. In all other cases the stop will occur immediately on completion of the order causing the
overflow.,

The Stop on Overflow key is not present on Pegasus 1.

25 -

Chapter 3

The Order— Code

This chapter is concerned with all the orders in the Pegasus order-code with the exception of
those in groups 0, 1 and 4, which are described in Chapter 2, and a few less commonly used orders, which
are described in later chapters^

3. 1 Multiplication

In Pegasus most of the orders of group 2 are concerned with multiplication and division; the
structure of this group is different from that of groups 0, Land 4.

If we multiply together two decimal numbers having each the same number of digits we get a product
which contains up to twice as many digits. For example,

381 X 615 = 234315
.381 X ,615 = .234315

The numbers on the left are the factors in the multiplication. In the same way if we multiply together
two binary fractions, each having 38 bits after the point, we shall get a product with 76 bits after
the point. If we add a sign-bit we get a 77-bit product. This requires two computer words for its
representation and a number of this kind is therefore called a double-length or double-precision number.
It is important that the computer should be able to form the full product of two words and generally
to manipulate double-length numbers. In Pegasus, accumulators 6 and 7 can be used in the usual way like
any of the other accumulators, but they have some special properties as well. They can be coupled
together to form b. double- length accumulator, in which double- length numbers can be conveniently handled.
The full double-length product is placed there when the computer performs a multiplication.

There are three multiplicat' jn orders, of which the simplest is the 20-order (read as "two-oh",
not twenty). Let us consider the action of the order

5.3 4 20

This order takes the number in ordinary register 5.3 and multiplies it by the number in A'4, the full
product being left in X6 and 7 (in place of the previous contents): the contents of 5.3 and 14 are
undisturbed. In carrying out this multiplication the machine of course takes note of the signs of the
two factors (i.e. the operands) and ensures that the product is correctly signed. The 20-order always
places the product in XG and 7, whatever the Af-address and A^- address written in the order; in fact the
computer is so built that the multiplication is actually carried out in accumulators 6 and 7 and not in
the mill at all. Let us now consider the way in which the product occupies the space available (78 bits)
in X6 and 7.

The product has. 77 binary digits, of which the left-most is the sign-bit. For convenience we shall
number these digits from the left: 0, 1, 2, ,,,, 76; so that digit is the sign-bit and digit 76 is the
least- significant bit of the product (this is an extension of the usual way of numbering the digits of a
single word - see Section 1.8), When the product is placed in X6 and 7, digits to 38 of the product
occupy X6 and the remaining digits (39 to 76) occupy XI, except that the sign-digit position in X7 is
not used and the digit there is always made zero. We can think of XG as receiving the left (or ms) half
of the product, including the sign-digit, whereas XI receives the right (or Is) half. This is illustrated
in Fig, 3.1, in which the digit-numbers are shown.

Digits 1 to 38
of product

sign-digit
of product

...

Digits 39 to 76
of product

always zero

Pig. 3.1. A double-length product in accumulators 6 and 7.

Since accumulators 6 and 7 are often used for special purposes it is convenient to introduce a
special notation for their contents. We shall write p for C(6) and q for C(7). The double-length number
formed by placing the digits of p and q side by side will be denoted by (pq). but when we write this we

- 26

THE ORDER-CODE 3.1

usually imply that the sign-digit in XI does not belong to the number and is zero (i.e. a > 0). With
the aid of this notation we can indicate the effect of the 20-order by means of the equation^

20 (pq) ' = n X X,

where (pq) ' means the double- length number after the order has been obeyed. The order can be
described as:

20 Multiply the content of the specified register by the content of the specified accumulator and
place the product in accumulators 6 and 7.

Just as with the orders of groups and 1, the yv-address can refer to any register (e.g. to an
accumulator).

If there are two numbers in 5.0 and 5.1 and we wish to find their full double-length product, we
could use the orders

5.0 3 00 First number to X3

5. 1 3 20 multiply by second number

One of the two factors must be in an accumulator, and we have here used X3. The product will be placed
automatically in X6 and 7 and we could, for example, store it in 5.2 and 5.3 by adding the following
extra orders

5.2 6 10 ms half of product to 5.2

5.3 7 10 Is half of product to 5.3.

A sequence of orders like this may be used whenever we wish to form the full product of two words; it
is unaffected by whether we interpret the operands as fractions or as integers.

Before going any further we must consider carefully how we are to interpret the double- length
number (pq). To illustrate these Interpretations we shall use products of 3-digit decimal numbers; and
to avoid complications we assume at first that all the numbers concerned are positive.

If we multiply two integers the product will be an integer, tt for example,

381. X 615. = 234315.

and if the two factors are fractions the product is a fraction:

.381 X .615 = .234315

and if one factor is an integer and the other a fraction then the product will be a mixed number with
its point in the middle:

381. X .615 = .381 X 615. = 234.315

In other words, if the point is to the left (or right) in both factors then it lies on the left (or
right) in the product; if the point is to the left in one factor and to the right in the other then it
lies in the middle in the product. All this applies to binary numbers just as well as to decimal ones;
it is illustrated diagrammatical ly in Pig. 3. 2, (page 28) where the boxes represent the digits and the
position of the binary point is marked.

The equation defining the effect of the 20-order can consequently be written

20 (pq)j = nj.Xj,

where the j suffixes indicate that the quantities are integers, or

20 (pq)p = rif..Xf.,

where the ^ suffixes indicate that the quantities are fractions. In other words, this means that the
double-length product is a fraction if both factors are fractions; it is an integer if both factors
are integers.

Frequently both factors are "small" positive integers, in which case the left half (i.e. p) of
the full product is zero. We can then take the right half (i.e. q) as the product (single-length).
Examination of the top line of Fig. 3.2 should help to clarify this. For example, suppose we have
two "small" positive integers in 5.0 and 5.1; we can place their product in the single register 3.2
by the following orders

5.0 6 00 first integer to X6

5. 1 6 20 multiply by second integer

3.2 7 10 product to 3.2 (X6 will be clear).

In this example X6 is used for two purposes; it first holds one of the numbers to be multiplied, but
this gets overwritten (i.e. replaced) by the left half of the product (which is in fact zero). A

t It is common practice to leave out multiplication signs or to replace them by points, so we shall
often write nx or n.x instead of n x ;e. But the special symbol (pq) does not, of course, mean the
product p "X- q.

tt We have written the decimal point in these integers merely to emphasize that it really lies to the
right of the Is digit.

- 27 -

3.1

THE ORDER-CODE

FACTORS

PRODUCT

Integer

Fraction

Integer

Fraction

Mixed Number

Fraction

Integer

Mixed Number

Pig. 3. 2 How the position of the binary point in a double' length product
is determined by its position in each of the two factors.

sequence of orders like this can be used whenever the product is non- negative and less than 2^^ (which
is about 250 000 million); for example, one factor could be as large as 250 000 and the other a million.
If the product is greater than or equal to 2^^ the above sequence of orders will not place the correct
product in 3.2, but only its 38 Is bits (such products cannot in fact be placed in a single register).
Alternatively, if the product is single- length but may be negative then steps must be taken to correct
the sign-bit in XI before the product is used; this will be explained in Section 3.8.

When we multiply two integers it is usually the right half only of the product which is of
interest. When we are dealing with fractions it is usually the left half that we wish to retain; this
should be clear after looking at the second line in Pig. 3.2. For example, if the two factors are 3/4 and
5/8 the product is 15/32, and all but the first few bits of the double- length product will be zero. As
a rule the digits appearing in X^ will not all be zero but they can often be neglected. We shall discuss
the multiplication of fractions in the next Section.

Let us now consider the multiplication of an Integer by a fraction. It should be clear from Pig. 3.2
that we must now regard the binary point in the product as lying between p and q. The integer in X6
(i.e. pj) after the multiplication is the integral partt of the product, and the fractional part is the
fraction in XI (i.e. qf). For example, suppose an integer a (= 9 say) is stored in 4.0 and a fraction
y (= % say) is in 4.1; the following orders will place in 4.2 the integral part (= 6) of the product
a.y (= 6%).

4.0 6 00 a to X6

4..1 6 20 multiply by y

4.2 6 10 integral part [a.y] to 4.2

The fractional part (= %) of the product will be left in XI.

If we write (pq)in for the number in the double-length accumulator interpreted as a mid-point
number, we can get another form of the defining equation of the 20-order

(pq).

'F-*I'

t The integral part at a number is the greatest integer which does not exceed the number, e.g. the
integral part of 3.27 is 3 and the integral part of 2 is 2. Integral parts are sometimes denoted by
square brackets; thus we can write C-«I1 for the integral part of x, and

[l5i^] = 15. [e] = 6, [o] = 0. [%] = 0.

We must note carefully how the above definition applies to negative numbers; in fact

[-6] = -6, [-254] = -3. and [-15'^] = -16.

The fractional part is the number minus its integral part; it is never negative. For example, the
integral part of 3.456 is 3 and the fractional part is 0.456.

THE ORDER-CODE 3.2

Before describing the multiplication of fractions we must examine further the double- length number
(pq). Suppose we know the values of p and q, what is the value of (pq)? We shall for the moment adhere
to the fractional convention and write pf, qp and (pq)f for the fractions concerned. It is easy to
find the value of (pq)f by a simple extension of the rules given in Section 1.8 for evaluating a single-
length fraction. The sign-digit (if it is 1) contributes -1 to the value, and if digit k of the product
is 1 it contributes 2'^ (we must now consider k as running from 1 to 76). Now the total contribution
from all the digits in X6 is simply pf. When we consider qp as the right half of a double-length fraction
we see that each of its digits is 38 places further to the right than usual; and consequently the
contribution made by these digits to the double-length fraction is g^- x 2"^^, i.e. eq^.. The value to be
assigned to the double-length fraction is therefore

(pi)f

with qp ^ 0. If qp is "negative we can still assign the value pp + eqp to the double- length number. On
the integer convention the two halves of the number, pj and qj, are integers and the point in (pq)j is
76 places further to the right than in (pq)p. The value of (pq)j is therefore

(pq)j = 2^^pj + qj

on the integer convention. When the product is a mid-point number we can write it (pq)ff; clearly

(pq)lg = Pj + 9^.

3.2 Rounded multiplication

We have explained in the preceding section that when we multiply two fractions together it is
usually only the left half (i.e. p) of the product that is of interest. The error committed in
disregarding the right half entirely is always less than £, which is about 0.00000 00000 036. If this
is small compared with the value of the product we can often legitimately take the single- length
fraction pp (in X6) to be the product. It is important to realise that as a rule pp is only an
approximation to the product; it is true that it is nonnally a very good approximation (the error is
in fact only €qp), but it is nevertheless subject to bias. This is because q is non-negative„ which
means that pp may (exceptionally) be equal to the product but will normally be too small by any amount
up to e. This bias may lead to considerable error in the later stages of an extended calculation, and
to avoid it we must round the product.

This rounding process is automatically carried out by the computer if the 21-order is used for
multiplication instead of the 20-order. Thus if we have two fractions in registers 5.3 and 5.4 and we
wish to place their rounded product in 5.5, we can use the following orders

5.3 2 00 first fraction to X2

5.4 2 21 multiply by second fraction and round

5.5 6 10 rounded product to 5.5.

The 21-order may be described as follows:

21 hfultiply the fraction in the specified register by the fraction in the specified accumulator and
place the rounded product in.X6 (X7 receives the rest of the product).

The content of XI after this order is seldom wanted, we shall shortly explain just what it is. The
rounding can be thought of as putting into X6 the best single- length approximation to the true double-
length product. This approximation may be larger or smaller than the correct product but the error
never exceeds ± %e and is unbiased.

As an example, suppose we have three fractions x, y and z in registers 5.0, 5.1 and 5.2
respectively, and we wish to place in 5.3 the quantity x + yz. Since the result is to be a single-
length fraction it must be rounded to get the closest approximation. The following orders will do what
is required

5. 1 6 00 y to X6

5.2 6 21 yz (rounded) to J6

5.0 6 01 X + yz (rounded)

5.3 6 10 result to 5.3.

The addition of the single- length fraction x to the product cannot affect the rounding.

The 21-order is intended mainly for multiplying fractions; but it is sometimes useful when one
faptor is an integer and the other a fraction, when the product is a mid-point double- length number
(see Pig.3.2). That part of the product which appears in X6 will now be the integer closest to the
product; if a 20-order had been used it would have been the integral part of the product. For example,
suppose an integer a (= 9 say) is stored in 4. and a fraction y (= % say) is in 4. 1; the following
orders will place in 4.2 the nearest integer (7) to the product a.y (= 6%).

4. 6 00 a to 16

4. 1 6 21 multiply by y and round

4.2 6 10 nearest integer to a.y to 4,2

The reader should compare this example with a similar one in the previous Section.

29 -

3.3

THE ORDER-CODE

To explain how the rounding is done in a 21-order it Is simplest to consider the product as a mid-
point number. Suppose we have used a 20-order to produce the product (pg)^, i.e. a double-length number
with integral part pj and fractional part g^. This number is now to be adjusted so that C(6) will be
the Integer nearest in value to (/>g)j^. Clearly we need not alter pj if q„ < % and we should increase p
by 1 if qf^ ^ % This can be done by adding ^k to q^ and adding any carry which then occurs to the least-"^
significant digit of pj, i.e. we must add ^ to (,pq)^. This is shown in Pig. 3. 3 (adding 1 to digit 39 of
the product is equivalent to adding Hi to (pqr)^). This rounding is done automatically by the

sign-digit
of number

zero add one here

Pig. 3.3 Rounding a double- length fraction to produce a single-length
fraction in X6.

21-order. The order may thus be defined by means of the equation

21 ipq)^ = nj.xjf + ^ = np.Xj+^A (g'^0),

if the product of an integer and a fraction is being formed; or by means of the equation

21 (pq)l. = n^.jc^+^e (q'^0).

if we are multiplying two fractions. t

One of the reasons for providing special register 33, whose content is % (see Section 2.9), is to
facilitate the rounding of double- length fractions which are not obtained by a simple multiplication.
The rounding of a double- length number should be done only when the closest single- length approximation
to its left half is needed.

3.3 Cumulative multiplication

The 22-order is the third of the multiplication orders in the Pegasus order-code; it facilitates
the accumulation of products, which is a frequently needed operation. This order may be defined astt

(pg) ' = (pg) + n,x,

or verbally as follows:

22 Multiply the content of the specified register by the content of the specified accumulator and
add the resulting double- length product into the double-length accumulator ( X6 and 7).

In this order the factors may be integers or fractions provided (pg) and {pq) ' are properly interpreted.

Por example, suppose a, b, c, d are four positive integers stored in 3.0 to 3.3, and we have to
place the integer ab + cd in 3.4, we could use the following orders

3.0 5 00

3. 1 5 20

3.2 5 00

3.3 5 22

3.4 7 10

a to XZ

ab to XI

c to X5

add cd to ab in XI

ab + cd to 3.4.

As another illustration, suppose u, v, w, x, y, z are six fractions held in 5.0 to 5.5, and we
wish to evaluate the fraction uv + wa: + yz and place it in 5.6. In this case we want a rounded single-
length result and we must therefore arrange to add Hie to the sum of products before storing the ms
half; this is most easily done by using a 21-order to form the first product, since it does not
matter at which stage the addition of %£ is done.

t It is instructive to compare .this method of rounding with that applicable to decimal numbers.
Suppose we wish to discard the last 3 digits of the number 0.142857. The result we should retain
is 0.143, and this can be got by adding 0.0005 to the number before discarding the unwanted digits
The amount to be added is % in the last place to be retained.

tt Strictly speaking this equation should read

22 (pg) ' = p + eg + n.x,
on the fractional convention, since there is no restriction on the sign of the original C(7).

30 -

THE ORDER-CODE 3.4

5.0 1 00 u to XI

5. 1 1 21 (pq)' - ui; + ^

5. 2 1 00 a; to Jl

5.3 1 22 add wx to (pq)

5.4 1 00 y to XI

5.5 1 22 add yz

5.6 6 10 rounded result to 5,6.

Sequences of orders like this are useful in the evaluation of the scalar product of two vectors.

It should be noted that the three multiplication orders are all correctly signed; they will give
arithmetically correct results regardless of the signs of the operands. This of course assumes that we
require either the full double-length product or the single-length rounded product in X6. If the right
half of the product (in XI) is all that is wanted, as when multiplying "small" integers, special
arrangements (described in Sections 3.8 and 5.10) will have to be made if the result is likely to be
negative; this does not often occur.

Overflow (see Section 2.11) may occur as a result of a multiplication. The 20 and 21 orders can
cause overflow only if the fraction -1.0 is squared, the true result should be +1.0 (or +1.0 + !4e for the
21-order) which is outside the permitted range; the result actually obtained in this case will be -1.0
(or -1.0 +1^ for the 21-order) and OVR will be set. With the 22-order there are of course more
possibilities for overflow; but it should be noted that it is only the final result which determines
whether OVR gets set, i.e. overflow will not occur if -1.0 is squared and added to a negative number
standing in the double- length accumulator.

Multiplication is, not unnaturally, a slower operation than addition or subtraction. A single
order of groups 0, 1 or 4 is obeyed in 3 word-times if it is an a-order, or in 2 word-times if it is a
6-order (see Section 2.4). A 20- or 21-order takes 13 extra word-times, and a 22-order takes 14 extra
word-times. Thus a 20-order takes altogether 16 word-times if it is an a-order, or 15 if it is a 6-
order. As a rough figure one can say that multiplication takes about 2 milliseconds. In timing a
section of programme containing multiplication orders it is usually simplest to find the time on the
assumption that multiplication is as fast as addition, and one can then add in 13 word-times. for each
20- and 21-order and 14 word-times for each 22-order.

3.4 Double-length addition and subtraction

The numbers normally handled by Pegasus are represented by single words and have 39 binary digits.
We have seen how double- length numbers can arise as a result of multiplications. It is sometimes
desirable to operate entirely with double-length numbers, which contain the equivalent of nearly 23
decimal digits. When adding or subtracting double- length numbers we must arrange for carries to take
place between the two words. The 23-order is provided to facilitate this; it is called the justify
order.

Operations on double-length numbers are usually carried out in the double-length accumulator formed
by X6 and XI. A fraction in this accumulator is denoted by p + €q in general, but the notation (pq) is
preferred when q is non-negative. It is usually desirable to adjust a double-length number so that its
right (or Is) half is non-negative (this is essential when shifting, as will be explained in Section 3.7),
such a number is said to be in standard form.

Let us consider the addition of a double-length number in registers 5.0 and 5.1 to another such
number in X6 and 7; we need not assume that these numbers are in standard form. We must first add the
right halves:

5.1 7 01

This is the operation which may give rise to a carry which is to be added to the Is end of the left half
of the sum. A carry will be necessary if the above order sets OVR, or if C(7) is negative. The justify
order therefore examines OVR and the sign bit in J7, determines from them, and effects, the required carry,
and then clears OVR and the sign bit in XT. There are the following four possibilities.

OVR

Sign

C(7)

Carry required

clear

-€

set

-26

set

This table may easily be checked by noting that if OVR is set then the sign of C(7) is incorrect. The
23-order adds the carry into the Is end of the register specified by its iV- address (the J- address is
not used and we usually write zero here). The complete sequence for adding the above two double- length
numbers is therefore as follows.

5.1 7 01 add right halves
6 23 justify into X6
5.0 6 01 add left halves.

Subtraction may be done by replacing the two 01-orders in this sequence by 03-orders.

3.5 THE ORDER-CODE

A verbal description of the 23-order reads as follows:

23 Put into standard form the double- length number in the specified register and X7, on the

assumption that a preceding addition or subtraction in X7 has determined C(l) and the state
of OVR. Leave C(7) non-negative, and leave OVR clear unless the left half of the number
(in the register) overflows. *

The 23-order takes the same time as an OO-order.

The operation of the order may be defined algebraically as follows:

23 (nq)' = n + eg + contribution from OVR, (q' ^ 0).

where (nq)' is the double-length fraction in N and XI with g' ^ 0, and the "contribution from OVR" is
zero if OVR is clear but is ± 26 if OVR is set (the sign being opposite to that of q); OVR is left
clear unless n' overflows.

The sequence given above for the addition of two double-length numbers will as a rule leave OVR
set only if the double-length sum exceeds capacity. There are exceptional cases in which the carry
causes overflow in the intermediate partial sum while the complete sum (after the last addition) is
within capacitytt.

The 23-order may also be used when operating with numbers more than two words long (multi- length
numbers).

3.5 Division

Division and multiplication are normally thought of as inverse processes and it is desirable that
the correspondence between them should be reflected in the way the computer operates. We should therefore
expect that if we divide the product of two numbers by one of them, then the resulting quotient should
be the other number; or, in symbols, we expect that

o. b _ ,
a

One consequence of this apparently trivial statement is that, since the product a. 6 is a double-length
number, we must be prepared to divide a double- length number by a single-length number. It is also
useful on many occasions to get not only the quotient but also the remainder after a division (in the
above example the remainder is zero). In general, therefore, we require to take a double- length
dividend and a single-length divisor and from them to find a quotient and remainder, which will both be
single- length numbers, and we must provide for negative numbers as well as positive ones.

To take a simple example, consider the division of 43 by 5. Here 43 is the dividend and 5 is the
divisor; and we can see that 5 goes 8 times into 43 with 3 left over, so that the quotient is 8 and the
remainder is 3. We should note that the remainder is less than the divisor and is non-negative (i.e.
< 3 < 5), and there is a simple relationship between these numbers which can be written as

^ = 8 +-I- (0 ^-|< 1)

or, what amounts to the same thing.

5 5 ' " 5

43 = 5x8+3 (0^3<5)

The quotient can usefully be thought of as the integral part of the fraction 43/5. In general, if we
write u and v for the dividend and divisor, and q and r for the quotient and remainder, we can write

+ J1

(0 $-^< 1), (1)

u = v.q + r (0 ^ r < v). < (2)

These two statements are equivalent provided the divisor v is positive. Assuming that the divisor is
positive for the moment, either statement completely determines q and r when u and v are specified (it
is assumed that all these quantities are integers and that v is not zero).

The 24-order may be used to carry out this kind of division process. Let us consider the order

5.2 3 24

In this division order the double-length dividend is formed by taking C(3) for the left half and C(7)

t It is assumed throughout that iV t^ 7. The sign-bit in Xl will not be cleared by the order

7 23
tt FOr example, with the numbers

p = I -e, q = %; C(5.0) = -¥l, C(5.1) = Vi,
More elaborate sequences of orders can be written down for handling such numbers.

THE ORDER-CODE 3.5

for the right half. The divisor is C(5.2). Thus

C(3) C(7)

Dividend

C(5.2)
Divisor

The left half of the dividend is taken from the accumulator specified in the order; it is mjide into a
double- length number by adjoining the content of XI as shown above. If the dividend were in the double-
length accumulator (X6 and 7) we would simply specify XG in the division order. The divisor is the
content of the specified register. The 24-order always places the quotient in XI (the mnemonic "q for
quotient" may help in remembering this) and the remainder in X6.

For example, to divide a double- length Integer u in X6 and 7 by a single- length integer v in 3.0,
placing the quotient g in 4.0 and the remainder r in 4.1, we could use the following sequence of orders,

3.0 6 24 divide u by u

4.0 7 10 quotient to 4.0

4. 1 6 10 remainder to 4. 1.

The quantities u, v, q, r satisfy the relationships given above.

Frequently the dividend is a single-length integer, when we put it into XI and specify XO in the
division order; the left half of the double- length dividend is then zero. For example, to find the
quotient and remainder when 43 is divided by 5 we could use the orders

43) 7 40 43 to XI

(?) 6 40 5 to J6

6 24

Here the dividend is formed from C(0) and C(7), and the divisor is C(6); as always the quotient (in
this case 8) appears in XI and the remainder (3 here) in A'6. Note that the quotient and remainder always
replace the previous contents of XI and X6.

The double-length dividend in a 24-order may be denoted by (xq), since it is formed from x, the
content of the specified accumulator, and q, the content of XT. This notation strictly implies, however,
that g is not negative, and in fact there is no restriction on the sign of q; so that to be precise we
should write the dividend x,2^^ + q on the integer convention, or x + £q on the fractional convention.
Since this notation is clumsy we shall continue to write (xq) for the dividend, with the understanding
that in this case q may be negative. t

The operation of the 24-order may be described, on the integer convention, by either of the
relationships

Ifil = g' +_p1 (0 <Zl< 1). (3)

n n n

(xq) = n.q' + p' (0 ^ p' < n) , (4)

which the reader should compare with those numbered (1) and (2) at the beginning of this Section. Since
we must allow the divisor n to be negative we prefer the form (3) above (the inequality in (4) is
Impossible when n is negative). The definition of the 24-order on the integer convention is therefore
as follows: ft

24 g' +^ = i^ (0<lL< 1),
n n n

9
It may be described verbally as follows:

24 Divide the double'length number in the specified accumulator and X7 by the number in the
register; place the quotient in X7 and the remainder in X6.

These descriptions are valid even for negative divisors or dividends; note that the remainder (unless
it be zero) always has the same sign as the divisor. It is not often that negative remainders are of
interest. The quotient produced by the 24-order is always the integral part of the number (xq)/n, i.e.
the great6st integer not exceeding this number (see Section 3.1).

t At the start of the division process the divisor is placed in a special multiplicand/divisor register;
the left half (x) of the dividend is simultaneously copied into X6 and a partial justification occurs
in which q is made non-negative and C(6) correspondingly corrected (if necessary). During the
division the divisor gets added and subtracted into X6, the altered dividend is shifted up (double-
length), and the quotient is built up, digit by digit, in XT,

tt We adhere to the convention that the results of an order (p ' and q' in this case) appear on the left
in the defining equation.

- 33 -

3.6 THE ORDER-CODE

The diagram shows the five words concerned in the order.

Dividend

Divisor

p' in J^:6

Remainder

q' in XI

Quotient

The overflow-indicator will be set on attempting to divide by zero or if the quotient exceeds capacity.
The quotient can never exceed capacity if the dividend is a single-length integer (unless q = -2^8 and
n = -1).

As an illustration of the use of the 24-order, let us suppose that a positive integer stored in
5.0 represents a sum of money expressed in units of a penny, and that we require to convert it to
£. s. d. , placing the number of £' s in 4.0, the number of shillings in 4.1 and the number of pence in 4.2.
If we denote these last three quantities by I, s and d respectively, and p is the sum involved then

p = 240Z + 12.9 + d,

= 12(20i + s) + d,

so that d is the remainder when p is divided by 12 and 20Z + s is the quotient; if this quotient is
in turn divided by 20 we can find I and s. Thus the following sequence of orders can be used.

12 to xe

12) 6 40

5.0 7 00 p to XI

6 24 divide p by 12

4.2 6 10 d to 4.2

(20) 6 40

6 24

4.0 7 10

4.1 6 10

divide (201 + s) by 20
I to 4.0
s to 4.1.

3.6 Rounded division

We have so far considered the division of integers, where the 24-order takes a dividend (.xq)j and
a divisor n^ and evaluates a quotient qjr and remainder pjr connected by the relationships

.li =

(xq)

(0 <-^< 1)

(1)

The J suffixes indicate that the quantities are all integers. These relationships must be supplemented
by the information that all the numbers concerned are integers. If we wish to divide fractions we must
interpret all the words concerned as fractions and the above relationships must be written

(*g)i

(0 <-^< 1).

(2)

We must also supplement these relationships by the information that all the numbers concerned are
"fractions", in the sense that they are expressible by words according to the usual conventions; this
implies that they are all integer multiples of e.

In general if we express in binary the ratio of two numbers (xq) and n we obtain an infinitely
long number, which can be called their "true quotient". For example, if the dividend (xq) is 1/8 and
the divisor n is 7/8, the true quotient is 0.001001001001001... (this corresponds in decimal to the

fact that 0.1/0.7 = 0.142857142857 ). In the 24-order the division process is stopped when the

first word-full of this true quotient has been evaluated; and it is at this stage that we get a
remainder, which is the number we could use as a new dividend (with the original divisor) to continue
the division process. Apart from the exceptional case when the remainder p' is zero, the quotient q^
obtained in this way is an approximation which is always less than the true (infinitely long)

34 -

THE ORDER-CODE 3.6

quotient, t In fact the value of the true quotient always lies between q' and q' + e on the fractional
convention, and sometimes g' + e is a better approximation than qf^ When we are dividing fractions. it
is this best single-word approximation to the true quotient that is usually wanted; it never differs
from the true quotient by more than ^ and is unbiased. We here have a situation similar to that
arising in the 21-order; in both cases we wish to get the best single-word approximation to a long
number, and this is done by rounding the long number.

The 25-order is provided to give rounded quotients. It is generally similar to the 24-order but
it yields a quotient q' which never differs from the true quotient by more than %e. The 25-order may
be defined by the relationships

25 7',+ e^' = 1^ (^^lL<^), (4)

n n n

on the fractional convention. The only difference from the relationships (2) is in the inequality
satisfied by p'/n. As with the 24-order, there is no restriction on the sign of q in the dividend. A
verbal description is as follows:

25 Divide the double- length number in the specified accumulator and X7 by the number in the
register; place the rounded quotient in X7 (and the corresponding remainder in X6) .

When dealing with fractions the remainder is not often needed but p' is still a "true" remainder, in

the sense that it may be used as a new dividend to continue the division process; if this is done
the result will be a fraction numerically not exceeding ^A which may have either sign.

As an example, suppose u, v and ip are three fractions stored in 4.0, 4.1 and 4.2, and we wish to

place in 5.0 the fraction uv/w. Since we are dealing with fractions it is understood that we require

the rounded value. We can use a 20-order to evaluate the full double-length product uv since the
25-order can deal with this as dividend.

4.0 6 00 u to xe

4. 1 6 20 uv to X6 and 7

4.2 6 25 divide uv by v> (rounded)
5.0 7 10 quotient to 5.0.

In this example the double-length dividend is in X6 and 7. Note that the rounding is deferred as long
as possible.

Frequently the dividend, as well as the divisor, in a single-length fraction. Suppose, for example,
we have to divide a fraction in X3 by another fraction in 5.0 and place the rounded quotient in 5.1. We
could use the following orders:

7 00 clear right half of dividend

5.0 3 25 divide

5.1 7 10 quotient to 5.1.

This kind of division occurs so often in practical calculations that a special order, function 26, has
been provided for it. In the 26-order the dividend is simply the single- length fraction in the
specified accumulator, in other respects the order resembles the 25-order. Thus in the above example
the following orders provide a solution:

5.0 3 26 divide C(3) by C(5.0)

5. 1 7 10 quotient to 5. 1.

The 26-order may be defined by the relationships

26 g' + e-P- = -^ (-^<^<%), (5)

n n n

on the fractional convention (it is of little use with integers). A verbal description is:

26 Divide the fraction in the specified accumulator by the fraction in the register; place
the rounded quotient in X7 (and the corresponding remainder in X6).

The overflow- indicator (OVR) will be set by any of the three division orders if the divisor is
zero, or if the quotient exceeds capacity.

The 25-order is sometimes useful with integers, when its defining relations become

25 q' ^iL = 1^ (-%<^<y,).

The quotient g' is then the integer nearest in value to (xq)/n.

t In fact .

n 'k^ n + €— = — ^ n

Since <-L. < l.

q' + e_ii_ = _ir:i±< g' + e, (3)

- 35 -

3.7 THE ORDER-CODE

Suppose a and 6 are the dividend and divisor respectively in a division; the 24-order produces a

rounded-down approximation to a/b, i.e. on the integer convention the quotient given by the order is
the greatest Integer not exceeding a/b. The 25-order gives an approximation which is rounded to the

nearest integer (in case of ambiguity it rounds up). Occasionally we want a quotient which is rounded

up, i.e. it is the smallest integer not less than a/b. We can get this by dividing -a by 6 (or a by -6)

and changing the sign of the quotient. For example, if a and b are integers in 5.0 and 5.1 we can place
a rounded-up quotient in 4.0 by means of the orders: *

5.0 7 02 -a to XI

5.1 24 divide by b

4.0 7 12 minus quotient to 4.0.

A rounded-down quotient would be obtained by replacing the 02 and 12 orders by 00 and 10 orders
respectively.

It is sometimes useful to use a double-length dividend in which the binary point is between the
two halves. If such a number is divided by an integer then the quotient must be Interpreted as a
fraction, and vice versa. This fact can be derived either from the appropriate defining equations or
by consideration of the inverse process of multiplication. For example, suppose we have a fraction u
in 5.0 and we wish to replace it by 7u/13 we could use the following sequence of orders

@ 6 40 7 to Xe

5.0 6 20 lu to X6 and 7

(13) 5 40 13 to A'5

5 6 25 divide 7u by 13

5.0 7 10 rounded quotient to 5.0.
To divide a fraction u in 5.0 by 18 we could use the following orders

5.0 7 00
(18) 6 40

6 25

5.0 7 10

To place in 4.0 the integral part of u/v, where u and v are the fractions in 5.0 and 5.1 we could use
this sequence.

5.0 7 00

5. 1 24
4.0 7 10

Any division order takes a time of about 5fA milliseconds, or precisely 41 word- times longer than
an 00-order.

The 27- order is described in chapter 11.

3.7 Shifts, the orders of group 5

The orders of group 5 are concerned with shifting the digits of words, i.e. taking the binary
digits of a word and moving them to the left or right. Let us consider the equivalent operation on a
decimal number, for example,

0.00123

If we move the decimal point two places to the right in this number we get 0. 123 (discarding extra
zeros), i.e. a number 10^ = IQO times as big as the original number. We prefer to think of this
operation as moving the digits of the number two places to the left (past the fixed decimal point).
In general, if we move, the digits of a decimal number N places to the left we shall have multiplied
the number by 10^; movement in the opposite direction corresponds to division by ION (or multiplication
by lO''*^). In binary the effects are similar but we must use powers of 2 rather than powers of 10.
For example, if we take the binary fraction 0.00101, which has the value 5/32, and shift its digits
two places to the left we get 0.101, which has the value 5/8 = 4 x 5/32 (since 2^ = 4). If we had
shifted the digits one place to the right we would have got 0.000101, whose value is 5/64 = ^^ x 5/32.
To prevent confusion we shall often talk of shifting a number up or down instead of shifting its
digits left or right. If we shift a number up it becomes numerically larger.

The 50-order is the order normally used for shifting a number up. For example, the order

5 50

causes the number in X5 to be shifted up 3 places, i.e. multiplied by 2^ = 8. The number of places
through which it is to be shifted is called the shift-number and is written in the ;V-address position
in the order; as with the orders of group 4, this number is encircled since it is not an address.

- 36

THE ORDER-CODE 3.7

The 51-order is written in a similar way and is used to shift numbers down. For example the order

will shift the number in X2 down six places, or divide it by 2^ = 64.
The 50- and 51-orders may be defined by the equations

50 x' = 2'^x,

51 x' = 2-^x = x/^:

or, verbally, as follows.

50 Multiply the content of the specified accumulator by ^, where N is the number written first
in the order.

51 Divide the content of the specified accumulator by ^ , where N is the number written first in
the order.

These descriptions and equations apply to both integers and fractions. The shift-number (N) may be
anything from to 127, inclusive; a shift of zero places does not, of course, affect the number being
shifted.

The 50- and 51-orders are collectively called the single-length arithmetical shifts since they can
be used for shifting single- length numbers. Apart from the movement of digits which these orders cause
there are a few other effects which have been provided in order to facilitate their use.

If the 50-order is used to shift up a non-zero number then overflow will occur if the shift-
number is sufficiently large; thus OVR will be set if ^x exceeds capacity, t For example, overflow
will occur if * = ^ and N > 1, or it x - -% and N ^ 2. The computer maintains the word- length of 39
bits by (a) discarding the N digits shifted up beyond the sign-digit position, and (b) by supplying N
extra zeros at the right-hand (least-significant) end of the word. The result of the shift is exactly
2^x provided overflow does not occur; if overflow does take place the result obtained differs from the
correct one by a multiple of 2 (on the fractional convention).

The 51-order cannot set OVR but it has two special effects. In order to keep the result arithmeti-
cally correct the sign-digit is repeated during the shift, i.e. N copies of the sign-digit are supplied
at the left-hand end of the word. For example, if the binary numbers

0.101000... and 1.011000... ,

whose values are 5/8 and -5/8 respectively, are shifted down two places by a 51-order we get

0.00101000... and 1.11011000...

respectively, having the values 5/32 and -5/32.

In general the result of a 51~-order will not be exactly «/2^ since significant digits may be
shifted out of the word at the right-hand (Is) end. In order to minimize the error the result is
rounded, tt The method adopted is to add %e (on the fractional convention) to the number before dis-
carding the unwanted digits; this is equivalent to adding € to the result if the first discarded
digit is a one. In this way the error does not exceed ± i^, in fact

-%e < x/2^ - x' < ^,

on the fractional convention. Consequently the result is identical with that obtained from a rounded
multiplication by the fraction 2"^, or from a rounded division by the integer 2'^.

For example, suppose we have two fractions u and v in 5.0 and 5.1, and we have to put the
fractions u/16, 4v and 5v in 4.0 to 4.2. jwe must do the shifting in one of the accumulators, say XS.

5.0 6 OO' u to X6

(4) 6 51 divide by 2"* = 16

4.0 6 10 u/16 to 4.0

5. 1 6 00 t; to 16

(2) 6 50 multiply by 2^ = 4

4.1 6 10 4t; to 4.1

5.1 6 01 add v

4.2 6 10 5v to 4.2

Note that the order

3 51

t To shift N places the computer shifts one place iV times. If at any stage the sign-digit is
changed then OVR is set. The shifts take place in the mill, and the computer counts the shifts
in the order-register.

ft The situation is analogous to that obtaining in multiplication and division; in all these cases
we have a number which is too long to be fitted into one word.

37 -

3.7 THE ORDER-CODE

has the effect of halving the content of X3, and the order

(T)3 50

is equivalent to the order

3 3 01.

which doubles C(3). The order

0) 3 50

has no effect.

The orders with functions 52 and 53 are called the logical shifts; they closely resemble the
orders 50 and 51 respectively and are written in a similar way. These orders simply shift the digits
of the word without any of the special effects of the 50- and 51-orders. They are not primarily
intended for use on words representing numbers but on words used for special purposes which will be
described later. The 52-order is similar to the 50-order, the main difference being that OVR is not
affected. The 53-order is generally similar to the 51-order but there is no rounding and no
repetition of the sign-digit (N extra zeros are supplied at the left-hand end of the word). Thus the
orders

ClO) 4 52

10) 4 53

have the effect of replacing the 10 left-hand digits of C(4) by zeros. The 53-order may be used to
effect an unrounded arithmetical shift down provided the operand is non-negative.
The 52- and 53-orders may be described as follows.

52 Shift the binary digits of the word in the accumulator to the left (up) N places (N being
written in the first position in the order). Discard the N digits which are shifted beyond
the ms position; make the last N digits of the word all zeros. Do not affect OVR.

53 Shift the binary digits of the word in the accumulator to the right (down) N places (N being
written in the first position in the order). Discard the N digits which are shifted beyond the
Is position; make the first N digits of the word all zeros.

The orders with functions 54 and 55 are the double-length arithmetical shifts. The shifting
always occurs in the double- length accumulator formed from X6 and 7; the A^-address in the order is
not used by the computer and we usually write zero here. Thus the order

(e) 54

multiplies the double-length number (pq) by 2^ = 64, and the order

(T) 55

divides it by 2^ = 4. Since these shift orders are intended primarily for use with numbers, the overflow-
indicator may be set by a 54-order, and the sign-digit is repeated with the 55-order. There is,
however, no rounding when shifting down with a 55-order. t The double- length shift orders may be
defined by the equations

54 (pq)' = 2^(pq).

55 (pq)' = 2"^(pg) = (pq)/^ (unrounded).
A verbal description of the 54-order is as follows.

54 Multiply the double- length number in X6 and 7 by 2^, where N is the number written first in
the order.

A description of the 55-order can be obtained by changing the first word to "divide". These equations
and descriptions are valid on either the integer or the fractional conventions.

As an example, suppose three fractions u, v, w are stored in 5.0, 5.1 and 5.2. and we have to
place in 4.0 the fraction 2uv/w, We first form uv. double it with a 54-order and then divide by w.

5.0 6 00 u to X6

5.1 6 20 uv
(T) 54 2uv

5.2 6 25 divide by w (rounded)
4.0 7 10 2uv/w to 4.0.

It is important to note that q, the content of XT. must not be negative if arithmetically correct
results are required. It is only with the double-length shift orders (54 and 55) that we have to be

t The main reason for this is the absence of any double-length logical shifts, so that the 54- and
55-order8 are often used for logical purposes, where rounding would be very inconvenient.

THE ORDER-CODE

3.8

careful about the sign of C(7); if there is any doubt then the order

6 23

(see Section 3.4)

should be inserted immediately before the shift. It is only rarely that this need concern the programmer
since double- length numbers commonly occur only as a result of multiplication; they then always have a
non- negative right half.

In the two double- length shift orders the sign-digit in XI does not take part in the shift and is
made zero before any shifting occurs. t This digit is by-passed during the shift, so that digit 38 of
the double-length number (digit 38 in X6) can be thought of as lying immediately to the left of digit 39
(i.e. digit 1 in XI). This is shown in Pig. 3. 4.

may set
OVR 4-

3 4

sign-digit
of number

extra
zeros

zero

The action of the 54-order. Digits shifted up from XI hop over the sign-digit
in J7 (which is always zero), and appear at the right end of X6.

•■

_y \. -

— 1»

digits

shifted

off

(No rounding)

copies
of

sign-
digit

'sii
of

tn-c
nur

lig
nbe]

zerc

The action of the 55-order. Digits shifted down from ^6 hop over the sign-digit in
XI (which is always zero) and appear in the next position in XT.

Pig. 3. 4 The double- length arithmetical shifts.

Since in all the above shift orders the computer operates by repeatedly shifting one place at a
time, it follows that a long shift (of many places) is slower than a short shift. In fact the time
required by almost any order 50 to 55 to shift N places is just N word-times longer than a simple order
such as an 00-order. Por example the order

(lO) 3 51

requires 10 extra word-times. There is an exception in the order.

25) X 52

which takes no longer than a simple OO-order. This fast 25-place logical shift up (or Counter-to-
Modifier Shift) has a particular application when the technique of modification is used, and is
described further in Section 5.5.+ In a 52-order, where the shift is of N places, A^ ^ 25, there will
be a slow shift of A'-25 places followed by a fast shift of the remaining 25 places. In this case the
order will take N-25 extra word-times, for N < 25, the order takes iV extra word-times.

Most computers are provided with shift orders because they require little extra equipment and are
very useful. Since Pegasus is a binary machine the arithmetical shifts correspond to multiplication
and division by powers of 2; and it is perhaps when using a shift order that one is most conscious of
the binary nature of the machine' s operations. Apart from this numbers often have to be doubled or
halved. Longer shifts are very useful if the scaling of numbers is done in powers of 2. In addition
to their purely arithmetical applications, shifts are indispensable for the so-called logical operations,
in which words are treated as strings of binary digits which may have no numerical significance but
which are used to represent all kinds of information.

As an aid to remembering the shift orders we may note that the even orders (50,52,54) shift up
and the odd orders (51,53,55) shift down. The 56-order is called the normalize order and is described
in Section 9.1. The 57-order is described in chapter 11.

3.8 Jumps, the orders of group 6

The vital role of jump orders (sometimes called test, discimination, control transfer or branch
orders) in even very simple programmes should be clear from the illustrations of Section 1.5. In a

t The sign-digit of C(7) is not cleared by a null double-length shift, i.e. one in which the shift-
number is zero, such as

(o) 54

or (0) 55.

After such an order C(6) and C(7) are unaltered.

tt This shift order is not fast in Pegasus 1.

3.8

THE ORDER-CODE

typical calculation there are many points at which a choice has to be made between a number of alter-
native procedures; which one of these is selected depends on results obtained earlier. A multiple
choice can always be broken down into a number of simple two-way alternatives and the selection at
each of these can be made to depend on a very simple yes-no attribute of some number, such as its sign
(negative or not) or its zero-ness (I.e. whether it is zero or not). In Pegasus the orders are
normally obeyed sequentially and the Jump orders provide the possibility of interrupting the regular
sequence and causing the computer to start obeying orders elsewhere. This jump is usually conditional;
the content of one of the accumulators may be tested in a certain way and the Jump occurs if the test
is satisfied, if it is not the computer carries on with the next order as though the jump order had not
been there.

The first four orders of group 6 may be used to test the content of any accumulator. Consider,
for example, the order

0.7 5 60

which tests C(5). This order will cause a jump if C(5) is zero, in which case the computer will start
obeying orders from the a-order in 0.7. As an Illustration, let us suppose we are in the middle of a
programme and we have two alternative processes to be followed; the second of which Is to be chosen only
if a previously calculated number stored in 4.5 is zero. The following sequence shows how this might
be written.

0.0

0.1

4.5 6 00

2.0 6 60

5.0 4 00

5.1 4 21

I.-*

@ 6 40

2.0

5.0 6 20

I I
I I

number to be tested to J6

jump if C(6) =

First alternative process

Second alternative process

The number to be tested must first be put into one of the accumulators, here Jir6 is used. The
6-order in 0.0 will cause a jump to the a-order in 2.0 if C(6) is zero, and in this case the computer
obeys orders sequentially from 2.0 onwards. If C(6) is not zero no jump occurs and the computer
carries on with the a-order in 0. 1 as usual. It is customary to draw an arrow, as in the above
illustration, to show the path of a jump; these arrows are very helpful when one is studying a
programme.

We may wish to jump to a b-order instead of an a-order; in this case we simply write a + sign
after the address. For example, the order

2.0+ 4 60

will cause a jump to the fe-order in 2.0 if C(4) = 0. This way of writing addresses is used with all
the orders of group 6, and is not normally used with any other orders.
The first four orders of group 6 may be summarized as follows.

60 Jump to ;V if * =

61 Jump to /V if % i<^

62 Jump to ;V if X ^

63 Jump to ;V if X <

(jump if number in accumulator is zero)

(jump if number is not zero)

(jump if positive or zero)

(jump if negative).

In this description N stands for the a-order or 6-order specified in the way described above. The
jump may be to any order held in one of the ordinary registers in the computing store; there is no
provision for obeying orders anywhere else (e.g. in the accumulators or the main store).

It will be noted that these orders can be grouped into pairs with contrary sense (60 and 61;
62 and 63). If, for example, the order 0.2 7 60 causes a jump under certain circumstances then the
order 0.2 7 61 will, in the same circumstances, not cause a Jump, and vice versa. This pairing of
the jump orders is a great convenience to the programmer. The 62- and 63-orders test the sign- bit
only;

As an example, suppose we have an integer in 5.0 which should be 13; if it is 13 we take no
special action and carry on with the programme; if, on the other hand, the integer is not 13 we wish
to obey a special sequence of orders starting at the 6-order in 3.3.

THE ORDER-CODE

3.8

•►

0.0

5.0 7 00
(is) 7 43

3.3+ 7 61

integer to XI

subtract 13

jump if not zero (i.e. if integer }^ 13)

to 6-order
in 3.3.

As another example, suppose we have a number z in 5. 1 which may be negative, and if so we have to
replace it by its absolute value (-z, which will of course be positive).*

0.0

1
V — 1^

5.1 1 00
0.1+ 1 62
5. 1 1 12

2 to XI

jump if z is non-negative

replace z by -z if z is negative

Sometimes we want to jump unconditionally; there are several ways of doing this but the one
conventionally used is to test XO to see if its content is zero, which it always is, of course. Thus
to jump unconditionally to the a-order in 3.0 we would write

3.0 60

and this order would be underlined on the programme sheet to emphasize the break in the sequence of
orders. Such unconditional jumps may be used, for example, to prevent the computer from "obeying"
numbers or other constants.

Consider now the problem of finding the smallest prime factor of a positive whole number N. This
problem was discussed in Section 1.5 (Example (C)) and a flow-diagram of a possible process is given in
Pig. 1.4, In order to write a programme corresponding to this flow-diagram let us assume that N is
stored in 5.0; we shall use X5 to hold our trial divisor d. We ultimately require to print this and
stop; and, since we have not yet described how this can be done, we shall simply assume that the
necessary group of orders starts in 4.0 in the computing store.

START

0.0

y—

2
3

4
5

5.0

7 00

5 40
24

4.0

6 60

"(S

5 40

5 41

5.0

7 00

24
6 60

4.0

7 04

0.2+

7 63

5.0

5 00

4.0

test If
N is even
and set
d = 2

N to XI

set d = 2 in X5

divide yv by 2

jump if remainder is zero,

set cf = 1 in J5

increase rf by 2

N to XI

divide JV by rf (quotient = q, say, to XI)

jump if remainder is zero

d - q to XI

jump if rf - q is negative (i.e. d < q)

set d = N in X5

unconditional jump
dummy order (=0 00)

Obeyed only
if N is prime

Print d in X5
and stop

t The absolute value of i is called the Modulus of z and is written \z\ . For example, the absolute
values of '/4 and --Vi are both ^ so we can write

|-i4l = ^. Clearly |7|

7. 1-121

= 12.

- 41 -

3.8

THE ORDER-CODE

It was pointed out in Section 3. 1 that the product of two integers can be put Into XI by using
a 20-order. This product will be correct provided it is not negative and does not exceed about 250 000
million. We can now show how negative products of "small" integers can be correctly obtained. The full
double-length product in X6 and 7 is always correct; but if the product is "small" and we take it
from XI we must insert the missing 1-digit in the sign-position in XT if the product is negative
(i.e. if C(6) is negative), but not if C(6) is positive or zero. Suppose we have two integers in 5.0
and 5.1, and we have to place in 4.0 their (possible negative) single-length product.

0.0

5.0 7 00

5.1 7 20
0.2 6 62

32 7 01

4.0 7 10

multiply two integers
as usual

jump if product is non- negative

if product is negative add in a "V
in sign-bit of XI

Special register 32 is used to provide the missing 1-digit. It will be seen that we have to insert two
extra orders if the product can be negative: by use of a trick which we shall describe later (Section
5. 10) we can do this with the aid of only one extra order.

As a further exanple of the use of jumps let us consider the flow-diagrwn of Pig. 1.3 (Example (B)
of Section 1.5); this programme is designed to read in 1000 positive numbers via the input equipment,
select the largest of them, print it and stop. Since we have not yet described how input and output
are done, we shall assume that the group of orders needed to read in a single number is in the
computing store, and that all we have to do to read in a number is to jump to 3.0: we shall suppose
further that this group of orders, which may be called the input subroutine (this subject will be
discussed in Chapter 6), places the number read in (x, say) into X6 and then jumps to 0.3. In the
same way we assume that in order to print a number we need only put it in J6 and jump to 4.0, where
the necessary group of orders (the output routine) starts; these orders cause the computer to stop
when it has completed the printing. The quantity y, which at each stage is the largest number read
in thus far, will be stored in X2. The integer c, which is used to count the numbers as they are read
in, will be stored in J3. We assume that these are not disturbed by the input subroutine.

START

0.0

Input 4-

Subroutine 1

Input

Subroutine 5

0.2 3 00

2 00

3.0 60

set c = 1000 (for counting) in X3
set y = in Jir2

jump to Input subroutine (x to X6)
dummy order (equivalent to 00)

form y - X in X6

Jump if positive or zero (i.e. it y ^ x)

OT y - (y - x) = X in X2 (i.e. replace y by x)

replace c by c - 1 ]

> count
enter Input subroutine if c 4= J

y to X6 ready for printing

jump to Output routine to print y and stop

dummy order

The a-orders in 0.1 and 0.6 are unconditional jumps. The 6-orders in 0.1 and 0.6 are dummy orders
inserted to make up the order-pairs; any orders would have done here but we have used the conventional
00 order (which may be written simply as 0). The two orders labelled "count" should be noted
particularly; the content of X3 is initially 1000 and these orders reduce it by 1 each time a number
is read in, the result being tested; eventually C(3) will be reduced to zero and the jump will not
occur. We shall see later that these two orders can be replaced by one order (with function 67).

When one is sketching out a sequence of orders which includes a number of jumps it is often
impossible to fill in immediately the ;V- addresses of some of the jump-orders; these can be left blank
and written in afterwards. The use of arrows to show the paths of the jumps is of especial value in
helping one to insert the correct addresses.

- 42 -

THE ORDER-CODE

3.8

The orders with functions 64 and 65 test the overflow- indicator (OVR).
clears OVR, even if no jump occurs; they may be defined as follows.

Either of these orders

64 Jump to iV if OVR is clear; clear OVR,

65 Jump to yv if OVR is set; clear OVR,

The J-digits of the order are not used and we usually write zero here.
For example, the order

3.6+ 65

will cause a jump to the fc-order in 3.6 if the overflow- indicator is set; after the order has been
obeyed OVR will be clear, whether the jump took place or not. If we except writing into the main store
(when the computer stops if OVR is set), the only orders affected by the setting of OVR are the two
jump orders just described and the justify order (function 23, see Section 3.4); any of these orders
will clear OVR. The overflow- indicator can be thought of as a very "sensitive" device which clears
itself whenever it is looked at. There is no other order which clears OVR, though it can of course be
set in a number of ways.

It is impossible to lay down any universally applicable course of action should overflow occur in
the course of a programme. Sometimes overflow will not matter, in which case OVR may be cleared by,
for example, a 64- or 65-order jumping to the next order. Sometimes it may be possible to adjust the
sizes of certain numbers and repeat a part of the programme. Sometimes the only sensible action is to
print some information and stop. The choice of a suitable course of action is bound to be considerably
affected by the details of the particular problem.

The orders with functions 66 and 67 are used mainly in connection with the facility of modifying
orders and will be described in detail later. The 67-order has, however, other uses and we shall
therefore give now a brief description of it which is not quite precise. The 67-order is called the
unit- count order; roughly, its effect is to subtract one from a count-number, or counter, in the
specified accumulator and then test the result; if this is not zero a jump occurs to the order
specified in the iV-address. For example, suppose we have a small group of orders which have to be
obeyed a definite number of times, say 9 times. The relevant part of the prograirane could be written
as follows.

START ► (9) 2 40 set counter = 9 in J2

Loop or cycle of
►orders, to be
obeyed 9 times

Unit-count in X2

Here the counter in X2 is initially set equal to 9, just before entering the cycle of orders which have
to be obeyed 9 times. At the end of the cycle is a 67-order which reduces the counter by one at each
repetition, and causes a jump back to the beginning of the cycle until the counter has been reduced to
zero, when no jump takes place and the computer carries on with the next stage of the programme. On
examination it will be found that the cycle will be traversed exactly nine times. Of course, any
accumulator (other than JO) may be used for counting in this way provided its content is not disturbed
during the cycle (or, of course, the counter can be temporarily stored elsewhere at the beginning of
the cycle and reset in X2 just before the end).

The time taken by any jump order is usually the same as the time for a simple 00-order, i.e. 3
word-times if it is an a-order or 2 if it is a 6-order, whether the order causes a jump or not. If
however there is a jump to a 6-order (i.e. the address in the order ends with a + sign and the jump
actually occurs) then the computer obeys a dummy o-order Just before obeying the 6-order, such a jump
therefore takes 3 extra word- times. For example, consider the sequence:

0.0

— ^

1.2

(9) 2 40
0.0+ 2 67

0.0

5.0 5 00
0.3 5 61

▼

Here the two orders shown take a total of 5 word- times, whether or not the jump occurs. In the
following example there is a jump to a 6-order; if the jump takes place the total time is 8 word-times,
if there is no jump the time is 23 word-times (N.B. a 21-order needs 13 extra word-times).

0. 1+ 3 63
3 6 02
3 6 21

5.1 6 10

0.0

A In a loop which is traversed many times, jumps to a-orders are to be preferred.

43 -

3.9 THE ORDER-CODE

3.9 Stopping the computer

We may wish to stop the computer for any one of a number of reasons. For example, the machine may
have reached the end of the programme, or we may wish to change the tape in one of the tape-readers or
the setting of the handswitches, or perhaps an error has been detected (e.g. unexpected overflow, or a
mis-punched tape).

The computer can be stopped manually at any moment by means of one of the keys (or switches) on the
control panel. This key is called the Bun key, or sometimes, the Stop/Run key (it is visible near the
centre of Plate 11); it has three positions. When the key is up, in the position labelled RlIN, the
computer obeys orders successively in the usual way. When the key is placed in the middle position,
which is labelled STOP, the computer finishes the order it is currently obeying and then stops; if the
key is returned to the RUN position the machine will continue with the next order as usual. If the key
is pushed down from the STOP position into the position labelled SINGLE SHOT and then released, it will
spring back into the STOP position and the computer will have obeyed a single order; by doing this
repeatedly the computer will obey the orders of a programme one at a time, t

There are a number of orders which can stop the machine. We have already mentioned the fact that
the computer stops if trying to obey an order causing writing into the main store when OVR is set (this
is called a writing-with- overflow stop). An unassigned order also causes a stop. There are special
lights on the control panel (Plate 11) to indicate these two kinds of stop (and some others), and there
is a hooter which can be turned on as an additional warning (the key for this is labelled HOOT ON STOP
and is mounted on the left of the control panel).

Another order which stops the computer is a jump order which jumps to itself, for example:

eel

0.4

0.4 60

When this order is encountered the computer will take its next order from 0.4, i.e. the order will be
repeatedly obeyed. It Is reasonable to say that the machine has been stopped by this order since there
is no change in any stored word. A stop of this kind is called a loop stop (or dynamic stop); the
only way of getting the computer out of the loop is by operating the Start key (this will be described
later). Conditional loop- stops are useful; for example, the following pair of orders

' ►

0.0

will stop the computer only if C(5,0) is negative. Such loop stops are often used to detect errors in
the input data (e.g. punching errors or numbers that have the wrong sign). Any of the orders 60 to 64
(but not 65) can be used to produce a loop stop. An unconditional loop stop may be used to terminate a
complete programme.

Although there is no warning light associated with a loop stop, the fact that the computer has
stopped is usually immediately evident. Input and output cease and the monitors show a static display
in all registers. In fact the machine seems to be "dead".

Frequently we wish the computer to stop temporarily at a certain point in the programme, for
example to allow us to change an input tape, after which we want to allow the machine to continue. One
way of doing this is to use the 77- order. In this order only the function digits matter, so it is is
usually written:

When obeyed it stops the computer and lights one of the special warning lamps on the control panel
(and sounds the hooter if it is on). The machine can be caused to continue with the next order by
moving the Run key to STOP and then back to RUN; this process is called operating the Run key. The
stop caused by this order is essentially a temporary one; it is usually called a 77- stop.

Another kind of stop is the optional stop. In Section 2.6 we described how the 39 bits of an
order-pair are allocated; 19 bits are used for each order. The bit corresponding to the sign-digit
in a number-word belongs to neither of the orders; it is called the stop/go digit, and can be used to
stop the computer. It is usual for this digit to be a one, in which case the order-pair is called a
go order-pair; the computer obeys go order-pairs in the usual way as has been described. If the
stop/go digit is a zero then the order-pair is called a stop order-pair and the computer will normally
stop just before obeying either of the orders. When it stops the order-pair will just have entered the
order- register. There is a key on the control panel labelled INHIBIT OPTIONAL STOP; when this key is
down the computer will disregard the stop/ go digit and will treat all order-pairs as go order-pairs.
This is why the stop is described as optional. When the computer has stopped optionally in this way it
can be made to continue by operating the Run key as described above for the 77-order. There is also a
special light (and the hooter) to indicate the stop.

It is usual, when a programme is being developed, i.e. when it is being tested and made to work
for most of the order-pairs to be go order-pairs; here and there will be a few stop order-pairs at
strategic points. When the computer stops the monitors can be used to check the contents of various
registers, or a specially-prepared checking programjue can be called in to print out some information.

t This facility is intended primarily for use by the maintenance engineers; but it is sometimes
useful when testing a new programme.

44 -

THE ORDER-CODE

3.9

When the programme Is knovm to be correct the optional stops can be inhibited or, alternatively, all
the stop order-pairs can be changed into go order-pairs (this is easy to do by using certain special
features of the Initial Orders).

On the programme sheet go order- pairs are not specially indicated. A stop order-pair is indicated
by writing a full-stop after either the a-order or the fa-order (but not both). This full stop must be
written after the Af-address (modifier address) of the order; if one of the orders is modified the full-
stop would be written after it; if neither of the orders is modified the full-stop is usually written
after the a-order (it is customary, and advisable, to write a zero in the W- address of the order in such
a case). For example, here are two stop order-pairs:

5.0 6 00

0.0

4.0 6 21 4.

3.0 6 01 0.

6 6 01

If the computer jumps to a stop order-pair then there will be an optional stop as usual, even if the
jump is to the h-order.

There are therefore the following three main ways of programming the computer to stop.

(a) Loop Stop. This is used when it is not desired to carry on with the next order; for example at
the end of the programme, or because an error has been detected which cannot be put right automatically
(e.g. tape punching error).

(b) 77- Stop. This is used when some manual operation may be needed before the computer is to carry on
with the next order; for example the handswitch- setting or the input tape may have to be changed. It
should not be used unless it is reasonable to continue the programme.

The writing-with-overflow stop and the unassigned-order stop normally occur because of a programming
error or incorrect data. Like the loop stop they can be cleared only by operating the Start key, which
is normally used to call in the Initial Orders to read in further tape.

The computer will also stop if an attempt is made to read tape when there is none in the tape-
reader or when the tape is incorrectly positioned (e.g. upside down); this is called an input busy stop
and has its own special warning light (but the hooter does not sound). The input busy light usually
flickers on and off during normal input. Apart from a manual stop (the Run key at STOP), all the
other kinds of stop are due to machine faults. For convenience the stops are tabulated in Table 3. 1;
some of the details will be described in later chapters. The hooter will sound only if it is switched
on and if the Run key is in the RUN position.

Kind of Stop

Loop
77- order
Optional

Writing-with-OVR
Unassigned order

Stop on Overflow

Run key at STOP
Paper tape busy

Parity failure
Magnetic tape failure

Magnetic tape busy
Card Reader

Card Reader

Card Reader
Card Reader

Reason

Programmed «
stop

Error in I
programme J
or data |

Programmed
indicator
or error

Operator

Operator or
computer fault

Computer fault

Programme,
operator or
equipment fault

Magazine

empty

Stacker full

Wreck

Warning
Light

Yes

Yes
Yes

Yes

No
Yes

Yes(2)
Yes(2)

Yes
Yes

Yes

Yes
Yes

Hooter

Yes

Yes
Yes

Yes

No
No

Yes
Yes

Action to continue

Start Key

Run key (or Start)

Start key
Start key

Run key (or Start)

Ensure that input tape is in
reader. If computer fault,
hand over to maintenance
engineer.

Hand over to maintenance
engineer.

Clear or repeat order or hand
over to maintenance engineer

Start key (or hand over to
maintenance engineer)

More cards

Clear stacker

Hand over to maintenance
engineer

Table 3,1 Summary of stops

45 -

3.10 THE ORDER-CODE

It Should be noted that if, due to a programming error, the computer starts to "obey" numbers
instead of orders then it will probably stop fairly quickly for one reason or another. In the majority
of programmes nearly all the numbers used are positive, so that they appear as stop order-pairs if
interpreted as orders; this is why the computer was so designed that a stop order-pair is shown by a
zero sign-bit. if the numbers are small and negative then the a-order will represent a 77- stop; also
many numbers will represent unassigned orders, so the computer is likely to stop when obeying numbers
even if optional stops are inhibited.

The "non-existent" ordinary registers 6.0 to 6.7 and 7.0 to 7.7 can all be regarded as storing
zero permanently (like the dummy accumulator XO and the unused special registers, see Section 2.9). The
number zero, if interpreted as an order-pair, is two dummy orders (0 00) and is a stop order-pair.
If the computer is obeying orders from U5 and "runs off the end" after obeying the 6-order in 5.7 it
will therefore immediately encounter an optional stop. If optional stops are inhibited the computer
will quickly run through all the dummy orders in the non-existent ordinary registers and will then
return to start obeying orders at the a-order in 0.0.

3. 10 Main-store transfers, the orders of group 7

There are four orders concerned with the transfer of words between the computing store and the
main store; these are the orders with functions 70 to 73 and are the only orders concerned with the
main store. The transfers may be done either in blocks of 8 words or one word at a time, and they may
take place in either direction. Since these transfers involve both stores the numbers written in the
N- and J- positions of the orders are interpreted in a special way, they are not simply the addresses of
registers and accumulators. In connection with computers the word transfer is used in the sense of
copying (or posting)', the place from which the information is taken is not cleared and its content
after the transfer will be the same as before. The original content of the place into which informa-
tion is transferred is simply lost and replaced by the new words.

Before describing the transfer orders a summary of the main store addressing system might be
helpful. The block- and-position form of the address of a storage location is the one usually used;
this is similar to the notation used for the ordinary registers in the computing store. A letter B is
often written in front of addresses in the main store to prevent confusion with those in the computing
store. The blocks in the main store are numbered BO to B1023, of which jBO to B895 are in the non-
isolated part of the store. The blocks in the computing store are referred to as UO to (/5.

The 70-order is the single-word read order; it causes the computer to read one word from a
specified location in the main store and to place a copy of it in accumulator 1. For example, the
order

34 6 70

places in Xi a copy of the word in B34.6. The block-number in the main store is written in the iV-part
of the order and the position-number is written in the A^-part. Because the N- and X-parts of the
order are now both parts of a single address it is usual to write a "box" round them; the above order
would therefore usually be written

34 6 70

The box is merely a visual aid on the programme sheet; it is not punched on the tape. The word read
is always placed in XI; this restriction is required because all the binary digits in the ;V- and
X-parts of the order are needed to specify the main store address. In fact there are only 10 bits in
these two parts of the order so that the address written can be from BO.O to B127.7 only. These
locations will be referred to as the first part of the main store (they are the first 1024 locations
of the 7168 in the non- isolated part). Access is obtained to the rest of the main store by means of
modified orders; these will be described later.

A location in the main store may, if desired, be specified by its decimal address instead of its
block- and-position address. Thus the location whose block-and-position address is 34.6 has a decimal
address of 34 x 8 + 6 = 278. This kind of address may be written in a single-word transfer order in
the iV-position; a minus sign is then written in the J-position, For example, the order

278

is equivalent to the order f34 6 | 70 and in fact these are merely two different ways of writing the
same order; they are both converted to the same internal form when read in by the Initial Orders. The
decimal address written in the order may not exceed 1023.

The 71-order is the single-aord write order; the address is written in the same way as in a
70-order. The transfer occurs in the reverse direction; for example, the order

12 4

causes a copy of the word in XI to be placed in £12. 4 in the main store. The previous content of
B12.4 is lost. As before, decimal addresses may be used, and locations in the first part only of the
store may be directly specified. This order causes the computer to stop, light a warning lamp (and
sound the hooter) if it is encountered with the overflow- indicator set; this stop occurs instead of
the writing operation.

The 70- and 71-orders may be described as follows.

70 (singU'Word read) Place in XI a copy of the word in the main store whose block-number is in

the N-part and position number in the X-part of the order.

t The blocks are numbered 60 to 5639 on the 4096-word store, of which BO to B5I1 are non- Isolated.

46 -

THE ORDER-CODE

3.10

71 (single-word write) Place a copy of the word in XI in the main store location whose block--

number is in the N-part and posit ion- number in the X-part of the order; but stop if OVR is set.

As an illustration suppose we wish to copy into B83.2 and B83.3 the words in U5.0 and U5.1
respectively, and copy into BO. 6 the word in the location whose decimal address is 981:

5.0

1 00

1 83

3 71

5.1

1 00

1 83

"3171

1991

J 70

"1171

C(5.0) to XI

transfer to B83.2

C(5. 1) to XI

transfer to B83.3

read word from 981 into XI

transfer to BO. 6

The two single-word transfer orders are useful for the occasional transfer of odd words. For
example, it may take a considerable amount of calculation to arrive at a single number, which could
then be transferred to the main- store by a 71-order until it is needed later. Before any transfer can
take place the appropriate location must be available; the computer will usually have to wait until the
drum has turned so that the location is under the read/write heads. This waiting time may be as long
16 milliseconds but is on the average about half this. It is usually more efficient to use the block-
transfer orders, and this is the normal practice.

For blocks 0-15 of the main store, a single-word transfer may take up to 1.25 milliseconds, but
an average of .75 milliseconds should be allowed.

The 72-order is the block^read order; it may be used to read any block in the main store and
place a copy of it in any block in the computing store. For example, the order

51 3 72

causes the eight words in B51 to be copied into the corresponding registers of (73. After obeying the
order the word in U3.0 will be a copy of the word in B51.0, that in m.l will be a copy of the word in
B51. 1, etc. The main store block-number is written in the yv-part of the order (as with the single-word
transfer orders); the Jf-part of the order specifies the block in the computing store. In order to
emphasize that the J-part of the order is not the address of an accumulator we usually write a "box"
round it on the programme sheet. The above order would therefore usually be written

51 ||]72

These boxes are very lielpful when one is scanning a programme sheet either to pick out block-
transfer orders or to find out when a certain accumulator is used.

The 73-order is the block-write order; it is generally similar to the 72-order but writes the
content of a computing store block into a main store block. Thus the order

24 [0]73

stores the eight words of UO in the corresponding locations in B24; the original words in B24 are
lost, but the contents of the registers 0.0 to 0.7 are unchanged. If OVR is set when a 73-order is
encountered the computer will stop instead of writing.
The 72- and 73-orders may be described as follows.

72 (block- read) Bead the block in the main store whose address (block-number) is in the N-part
of the order and place a copy of its contents in the computing store block specified by the
X-part of the order.

73 (block-write) Take the computing store block specified by the X-part of the order and write a
copy of its contents into the main store block specified by the N-part of the order.

With the 72- and 73-orders, as with the single-word transfers, access can be obtained directly only
to the first part of the main store (BO to B127); we shall describe later how the remaining blocks are
transferred by using modified orders.

The 73-order is used mainly for writing newly- computed numbers or other information into the main
store for later use. The 72-order has two principal uses; it is used to read in new data from the
main store which are required in a computation, and it is also used to read in fresh blocks of programme
as required. It is the latter, important, application that we shall now discuss a little further.

Suppose the computer is obeying orders in UO and that the next section of programme which we wish
the computer to obey is in the main store, in BIO say. This kind of situation arises frequently since
it is quite exceptional for a whole progranme to fit into the computing store. The following orders
can be used to read BIO into UO and start obeying the new orders at 0.0

0.6

10 [0|72
0.0 60

Read BIO into UO
jump to 0.0

- 47

3.10

THE ORDER-CODE

It Should be noted that these two orders must form an order-pair, i.e. they must be held in the same
register when obeyed (0.6 is used here for illustration). The reason for this is that the computer
normally operates by placing an order-pair in its order-register, and then obeying the two orders in
it successively before returning to extract another order-pair from an ordinary register (see Section
2.4). When the a-order of the above order-pair is obeyed it causes all the words in UQ to be over-
written by the new words from BIO, In particular C(0, 6) is overwritten, but the fe-order is still in
the order- register and is correctly obeyed as written, causing a jump to 0.0. Should we wish to read
in more than one block of programme the extra block-read orders can be written at the end of the old
block or at the beginning of the first new block; for example, in the above illustration the first
two orders of BIO could read

0.0

B 10

11 (T]72

12 |2|72

read Bll into Ul
read B12 into (72

The' main store block-number (BIO here) may be written in the small "box" printed on the programme sheets
at the top right corner of each block. It will be seen that the complete programme has to be divided
into blocks, at most six of which can be in the computing store at a time (as a general rule there will
be only three or four since some space will be needed for numbers). The main aim is to ensure that,
so far as possible, speed is not lost by including programme- transfers in the "inner loops'" of pro-
grammes, i.e. in those loops which are traversed many times and which would be fast if there were no
programme- transfers. If possible the division of a programme into blocks should be made logically so
that the break between two fairly distinct stages of a programme does not occur in the middle of a block.

Order-pairs consisting of a block-read and a Jump as Illustrated above occur frequently in most
programmes. Occasionally the Jump can be dispensed with and the computer can run-on into the new
block. Consider, for example, the following sequence.

j 1

0.7 2 00

0.3

«- —

18 072

0.0 6 03

1.0 3 10

read B18 into £/0

The 6-order in 0.3 causes the whole of UO to be overwritten, including C(0.4), so that when the computer
places the next order-pair (from 0.4) in its order- register this will not be the order-pair written
above but the new order-pair in 0.4 (i.e. that from B18.4). The above block read-order resembles an
unconditional Jump, inasmuch as the next orders on the programme sheet are not obeyed after it, and it
is therefore underlined on the programme sheet and marked like an unconditional Jump.

There are no ordinary registers with addresses 6.0 to 6.7 or 7.0 to 7.7, although such addresses
may be written in orders since there are enough binary digits to specify them. Like the unused special
registers (see Section 2.9) and the dummy accumulator JO, the non-existent ordinary registers always
hold zero. This fact can be put to good use if we require to clear a block in the main store (i.e.
replace its content by zeros). Thus the block-write order

62 [6]73

will transfer (76 to B62, i.e. it will clear B62. The corresponding 72-order has no effect.

If Ul is specified in a block- transfer order then the computer will transfer to or from the block
of accumulators. This is a very useful facility. At a certain stage in a programme it may be
convenient to transfer the contents of the accumulators to a block in the main store, do some auxiliary
calculation needing a number of accumulators, and finally to restore the original contents of the
accumulators by a block read-order. It is conventional to use BO to store the accumulators in this way.
Thus the order

CI! 73
transfers C(0) to BO.O, C(l) to BO. 1, ..

[7] 72

C(7) to BO. 7; and the order

sets all the accumulators from the contents of BO (e.g. Xl will contain the word from BO. 4; any word
sent to JO is, of course, lost).

Most programmes can be divided up into fairly well marked stages, and during one of these certain
blocks of numbers will probably be read into the computing store, used and replaced by fresh blocks of
numbers; there will also, as a rule, be a continual flow of numbers into the main store. Programme
(i.e. orders) and numbers are as a rule kept in the computing store only as long as they are wanted;
when no longer needed for some stage of a calculation, programme blocks are overwritten by new matter
and numbers are transferred to the main store (if they are wanted later). If overflow occurs this is
usually a symptom of a programming error (e.g. the scaling of numbers may be wrong or a certain
combination of circumstances may not have been anticipated), or it may be due to incorrect data or a

THE ORDER-CODE

3.10

mis-punched data-tape. Since block-write orders are usually frequent the computer will normally stop
fairly soon after overflow has occurred. This stop prevents the machine from carrying on with a,
perhaps extensive, calculation using wrong numbers; it may therefore prevent the waste of much valuable
computer time.

The block- transfer orders, 72 and 73, like the single-word transfers, can be obeyed only when the
drum is in the right position, and the computer may therefore be held up before the transfer can occur.
As a rule block- transfer orders are to be preferred to single-word transfers since eight times as much
information is transferred for the same waiting time. By counting word-times it is possible to find out
the exact angular position of the drum at any stage in a programme, and hence to determine the waiting
time before any particular word or block can be transferred. This Is, however, a laborious undertaking
and is not often worth doing. It is usually adequate to allow half a drum revolution (i.e. about 8
milliseconds) waiting time for an isolated transfer order; to this should be added about 1^ milli-
seconds (for a 72- or 73-order) to allow for the time of the actual transfer. Very often we wish to
transfer several consecutively-numbered blocks; in this case we can take the first transfer order to be
an isolated one and allow about ffk milliseconds for It. Subsequent transfers are rather faster though,
and we need allow only 3 milliseconds for each of them; this is because consecutively-numbered blocks
are arranged in a special way on the drum and become available fairly quickljr after the first of them
has been transferred (see Appendix 4). In fact the blocks are so arranged that there is always time
between the transfer of two consecutive blocks for four simple orders (e. g. 00- or 10-orders) to be
obeyed without losing any time. For example, the following sequence will require about 12'/^ milliseconds
(assuming it to start at a random moment).

— ^

0.0

19 (4]72

1.2 6 00

(D 6 41

3 6 03

1.7 6 10
20 (5]72

read B19

" four simple orders

i-ead B20

For blocks 0-15 of the main store, a block- transfer order takes 1.25 milliseconds and there is no
waiting time.

The remaining orders of group 7 are not concerned with the main store. We have already described
the 77-order (Section 3.9), which is the temporary stop order.
▼ The 74-order is the external- conditioning order which may be used to select a number of alternative
input and output devices or generally to control equipment external to the computer proper. In a
standard Pegasus installation the order is used to select which of the two tape-readers is to be used
(see Section 6.3). The seven iV-bits of the order are used to determine the settings of seven relays
(the external-conditioning relays), which perform the necessary switching; the right-hand (Is bit) is
used to select the tape- reader so that the order

® 74

switches in the second tape-reader in place of the main tape- reader; and the order

(O) 74

selects the main tape-reader. The J-digits of the order are not used. The state of the seven external-
conditioning relays is shown by a set of seven lights on the monitor panel. The 74-order takes 2 word-
times (i.e, the same as a simple order). t

A special sensing register, 24, is provided to hold in its least-significant end the N-digits
(subsequent to modification) of the 74-order last obeyed,. It is thus possible for the programmer to
ensure that previous selections of input and output devices can be taken into account during any new
setting, without having to record such settings by programme. t +

The use of the '74-order is further discussed in Chapter 10. The 75-order is unassigned and will
cause the computer to stop. The 76-order is used to transfer blocks of words between the computing
store and the magnetic tape, card or line printer buffer stores. For example, the order

2 |4]76

causes the contents of t/4 and magnetic tape buffer-store block 2 to be interchanged. This order takes
8 word-times longer than a simple order such as an OO-order (i.e, in all, about 1.3 milliseconds).
A The 76-order and its use are further described in Chapters 10 and 11*.

T On Pegasus 1, the 74-order is much slower, and an average of 40 milliseconds should be
allowed for it.

'T Note that register 24 is not present in Pegasus 1,

"^ In a Pegasus 1 without magnetic tape, the 76-order Is unassigned.

3.11 THE ORDER-CODE

3. 11 IjOgical operations

In a typical programme a high proportion of the orders will be those transferring infomation from
one part of the store to another, or doing such operations as counting, testing, reading or punching
tape, and generally ensuring that the useful arithmetical operations carried out are those appropriate
to the data. These operations are often called "administrative" "organisational" or "red tape"
operations, and frequently large sections of a programme will be concerned entirely with them. If,
Instead of programming the calculation for a digital computer, we were to specify it to the operator of
a desk machine, many of these organisational operations would not be mentioned; they would be either
unnecessary or implied. A computer programme is consequently much longer and more detailed than a sheet
of instructions for human use. In order to facilitate the organisational parts of a programme the
computer is equipped with a number of special orders. Certain of these are usually called logical
orders. This term is not well defined; some authorities would apply it to many of the organisational
orders, such as the jumps. But we shall restrict it to those orders where the numerical values of the
words taking part are only of secondary significance and where the words are primarily thought of
simply as strings of binary digits. We have already described the two logical shift orders, functions
52 and 53 (see Section 3.7); we shall now describe the remaining orders of groups 0.1 and 4, which were
not included in Sections 2.7 and 2.10. We shall illustrate some of the uses of these orders by examples.

The simplest and most useful of these orders are the collating or and orders, with functions 05, 15
and 45; they can be used to pick out binary digits or groups of digits in words. They each operate
on two words (operands) and produce one new word as the result; this resultant word has 1 digits only
in those binary positions where both of the operands have I' s. This collating operation is sometimes
called logical multiplication since the result may be got by a digit-by-digit multiplication. It is
best described by an example: the result of collating

u = 0.11110 00011 10011 00010 10101 01111 10110 110
with V = 0.11000 11000 10101 10100 00000 10110 11111 010
is w = 0.11000 00000 10001 00000 00000 00110 10110 010

A digit of w is 1 only where the corresponding digits of u and v are 1' s. The operation may be written
symbolically as

W = U & V.

Note that the operation is synnetrical, so that

u & V = V & u,

for any two words u and v. The operation is also associative, i.e., if x, y and z denote any three
words, then

X & (y & z) = (x & y) & z,

so that we can leave out the brackets; in fact the word

X & y & z

will have a 1 digit only where x and y and r all have I digits. In most of the actual applications one
of the operands will have a block of consecutive 1' s and will have 0' s elsewhere; the ordejr will then
pick out from the other operand the group of digits corresponding to the block of 1' s. For example,

u = 0.00000 00000 00000 00000 00000 11111 mil 000

and V = 1.10110 11011 10000 00110 10011 01101 00111 001

then u&v = 0.00000 00000 00000 00000 00000 01101 00111 000.

In this example 10 of the bits of v have been left unchanged, but the others have all been replaced by
O's. A word (like u in this example) which is used to pick out certain digits in other words is some-
times called a collating mask (or collating constant).

It is useful to introduce an abbreviated notation which can be used to write down collating masks
and the like. We shall write 0" or 1" to indicate a group of n 0' s or I's, respectfvely. The word u
above, consisting of 26 O's, 10 I's and 3 O's, can be denoted by O^^l^Oo^. A collating mask consisting
of 20 I's followed by 19 O's will be written l^ooi^.

The orders with functions 05, 15 and 45 are analogous to the other orders in their respective
groups and may be defined as follows.

F Effect Description

05 x" = x & n In the binary digital positions where the

word in the register has 1' s, leave the
digits in the accumulator unchanged;
elsewhere replace them by 0' s.

15 n' = n & X In the binary digital positions where the

word in the accumulator has 1' s, leave the
digits in the register unchanged;
elsewhere replace them by 0' s.

- 50

THE ORDER-CODE

3.11

F Effect Description

45 x' = X & N In the binary digital positions where the

integer written first in the order has 1' s,
leave the digits in the accumulator
unchanged; elsewhere replace them by
0' s (N.B. the result can have 1* s only in
the 7 right-hand digits).

None of these orders can cause overflow.

As an illustration suppose we wish to divide into two parts the word o in 5.0; the left half (6)
is to consist of the first 30 bits of a, and the right half (c) is to consist of the last 9 bits of a.
We need a collating mask having 30 I's followed by 9 0' s (i.e., l^OO^*)

1.11111 11111 11111 11111 mil 11110 00000 000

which has the numerical value -2~^^ on the fractional convention, or -2^ = -512 as an integer. Let us
assume that b and c (made up to complete words by adjoining enough 0' s) are to be put into 5.1 and 5.2
respectively.

0.0

-512

— fl.

5.0

6 00

0.0

6 05

5.1

6 10

5.0

6 04

.5.2

6 10

collating mask (l^^O^)

a to X6

replace Is 9 bits of a by zeros, result fc.

6 to 5. 1

a ~ b = c

c to 5.2

Alternatively the complementary collating mask (O^^i^, i.e. 30 0' s followed by 9 I's) could have been
used; this could be written on the programme sheet as the integer +511, since it is 2^ - 1. We shall
describe in the next section how other sorts of collating masks may be written.

As another illustration, suppose we have to jump to 3.0 if the integer (k) in 5.7 is odd,, Here
we wish to examine the last digit of k, which will be 1 only if k is odd.

0.0

<«

5,7 7 00

(T) 7 45

3.0 7 61

k to XI

leave last bit of k in XI

jump if last bit is 1

Some calculations require the manipulation of many small integers. Now a small integer does not
require many digits for its representation; for example if all the integers to be handled are known
to be between zero and 1000 then they may be represented by only 10 binary digits (because 2^^ = 1024 >
1000); if the integers lie between -50 and +50 then 7 bits may be used to represent them (there are
101 different integers and this is less than 2^ = 128). It is obviously wasteful of storage space to
use a whole 39-bit word for a single such integer and one may instead pack several integers into one
word. The collating orders can be used to unpack such a word into its component parts, or to pack up
several components into a word. The packing and unpacking operations take time of course, but this may
be more than counterbalanced by the more efficient use that can be made of such orders as block-transfers.

If a component of a packed word represents a signed number it is natural to use its left-hand bit
to Indicate the sign in much the same way as the ordinary sign-bit of a word, i.e. for positive (or
zero) and 1 for negative. This can easily be done, but it sometimes simplifies the packing and
unpacking operations if the sign-bit is reversed, i.e. it is in a negative number and 1 in a positive
one (or zero). This corresponds to adding a constant (a power of 2 in fact) to every component, so as
to ensure it is non-negative, and packing the components as unsigned; the constant is easily subtracted
during the unpacking process. For example, suppose an integer k lies in the range

-64 «J fe ^ 63.

so that it can be represented by 7 bits. For simplicity let us assume the integer is stored as

Jfe + 64 in the 7 Is bits of a word in 5.0; note that fe + 64 is non-negative. The following orders will

place k in .^6.

3. 11 THE ORDER-CODE

5.0 6 00

(ut) 6 45 jfe + 64 in X6

(m) 6 43 fe in XB.

As a rule, when unpacking the components of a word, we require to place each component at the Is
end (or, sometimes, the ms end) of a word. The logical shifts are useful for this (their use can
eliminate some collating), and they can also be used to shift the various components during the packing
process. The double- length arithmetical shifts (functions 54 and 55) can often be used for logical
purposes; though one should not forget that OVR may get set by a 54-order, that the sign-bit in X6 will
be repeated in a 55- order, and that the sign-bit in XI is cleared and does not take part in the shift
(see Section 3.7). When using double-length shifts we can often dispense entirely with collating orders.
For example, suppose that in 5.0 digits 1 to 9 and digits 10 to 14 represent two unsigned integers which
are to be unpacked and placed at the Is end in 4.0 and 4.1 respectively:

5.0 7 00

6 00

(V) 54 shift first integer into Is end of X6

4.0 6 10 first integer to 4.0
6 00

(T) 54 shift second integer into ^6

4. 1 6 10 second integer to 4. 1

A similar technique can be used for packing.

The packing of several items of information into a word is not, of course, restricted to
integers; the components of such a word may be interpreted in a variety of ways, and may be of
different length. Even single binary digits may sometimes be useful; for example, they can be used
to Indicate whether or not a certain event has occurred; the sign-bit of a word can easily be used in
this way since a digit in this position is always available (in special register 32) and its presence
or absence in a word can easily be sensed by a 62- or 63-order. Alphabetical information has often to
be stored; for example, we may wish to print names of persons or the headings to columns of numbers.
Since there are fewer than 32 letters in the alphabet, a letter can be represented by 5 bits (e.g.

A = 00001, B = 00010, C = 00011 Z = 11010), though there are some advantages In using 6 bits if

other thaii purely alphabetical information is to be included. In general, any information which is
to be represented inside the computer must be encoded in some way.

Another logical operation provided in Pegasus is the not' equivalent operation; this is perhaps
less useful than the collating process, but has some applications. The orders are those with functions
06, 16 and 46. As with the collating orders there are two operands which are combined, digit-by-digit,
to give a result; but in the not-equivalent pi'ocess the resultant word has a 1 only in those positions
where the corresponding bits of the operands differ. For example, the result of performing the not-
equivalent operation between

u = 0. 11110 00011 10011 00010 10101 01111 10110 110
and t; = 0. IIOOO 11000 10101 10100 00000 10110 Hill 010
is w = 0.00110 11011 00110 10110 10101 11001 01001 100

The process is sometimes called logical addition since the result may be got by a digit-by-digit
addition (or subtraction) with no carry. Where the digits of v are O's the digits of w are the same
as those of u; where the digits of v are 1' s the digits of u> are the reverse of those of u.
The operation may be written symbolically as

w = u ^ V

(read as "u not-equivalent v"). As with the and operation, this is symmetrical and associative, i.e.
if X, y and z are any three words then

X ^ y = y t- X,

and

X ^ (y i^ z) = (x ^ y) ^ z,

so that we can leave out the brackets. The word

X ^ y ^ z

will have a 1 digit where either all three operands have 1* s or just one of them has a 1 (i.e. where
an odd number of the operands have 1' s) .

52 -

THE ORDER-CODE

3.11

The relevant orders may be defined as follows.

Effect

Description

06 x' - X p n

In the binary digital positions where the
word in the register has 1* s, reverse the
digits in the accumulator; elsewhere leave
them unchanged.

In the binary digital positions where the
word in the accumulator has I's, reverse
the digits in the register; elsewhere
leave them unchanged.

x' = X p N In the binary digital positions where the

integer written first in the order has 1* s,
reverse the digits in the accumulator;
elsewhere leave them unchanged (N. B. at
most the 7 right-hand digits will be changed).

None of these orders can cause overflow.

One of the most straightforward applications of the not-equivalent operation is the comparison of
two words, u and v say, to determine whether or not they are equal. If u and v are equal, digit for
digit, then u ^ v will be zero. Suppose, for example, that u and v are in 5.0 and 5.1, and we wish to
jump to 2.0 if they are exactly equal.

5.0 6 00

5.1 6 06
2.0 6 60

u to X6

u p V to X6

jump if u = V

Of course we could have subtracted the two words, the result would again be zero only if the operands
were equal (even if they have no numerical significance); the disadvantage of subtracting is that
overflow may occur.

The order 32 4 06 will reverse the sign-bit in accumulator 4 and leave all the other bits

unchanged. Note that if this order, or, in fact, any not-equivalent order, ^ is obeyed twice then there

is no change in any word (in this respect such an order resembles an 04, 14 or 44 order, except that

these may cause overflow). This may be expressed symbolically as follows, if u and v denote any two
words,

u ^ (u ^ v) = V,

V ^ (u f^ v) = u.

This process is useful if we wish to interchange two words. Suppose, for example, that we have to
interchange u = C(6) and v = C(5.0).

5.0 6 06 u ^ V in Xe

5.0 6 16 V ^ (u ?^ y) = u in 5.0

5.0 6 06 u ^ (u ^ v) = y in J6

Interchanging can be done in other ways but this method has the advantages that no working space is
used and overflow cannot occur.

It is convenient to introduce a special notation to indicate certain other logical operations which
can be performed by combining several orders. We shall write to mean a word all of whose digits are
zero, and (-1) for the word whose digits are all I's (these are the values of the words as integers).
In the following, u, v and w represent any three words. Clearly

also

u & u = u, u(S:0 = 0, and u & (-1) = u,

u5^u = 0, U9^0 = u.

A bar over a letter will denote the not operation, i.e. the reversal of all the digits, so that

u 9^ (-1) =u, u<&u = 0, u 4u ^^ (-1).

For example, to reverse all the digits of C(5.0) we could use the orders

(-1) to XQ

Ul = {u ^ (~1)) in 5.0

exclude such orders as 4 4 06 of course.

- 53 -

3.11 THE ORDER-CODE

The and not operation can be progrwnmed by using either of the identities

u & V = (u & v) j^ a = (u ^ v) & u,

which can easily be proved; it is equivalent to collating with a complementary mask. If, for example,
u and V are in 5.0 and 5. 1 respectively, we can put u <& i; in ^6 by the orders

5.0 6 00 u

5. 1 6 05 u & V

5.0 6 06 a j^ (u & v) = u & V

The last order here could be replaced by an 04-order.

The or (or mix) operation is sometimes useful, it may be denoted by

u V v;

the resulting word has 1' s where either or both of the operands have 1' s. It can be programmed by using
one of the identities

u 1 V = (u -^ v) -~ {u & v) ,

UvV = (u & v) ^ V = {(u ^ v) & u} ^v = (u & v) ^ U ^ V,

which can be easily proved. For example, if u and v are in 5.0 and 5.1, the following orders will put
u ^ V in X6,

5.0 6 00 u

5.1 6 06 (u ji v)

5.0 6 05 (u ^ v) & u

5. 1 6 06 UVV

As another example, suppose u is in XG and v in 5.0, and we have to mix the last 5 bits of « into 5.0,
i.e. we wish to insert in the last 5 bits of C(5.0) any 1' s there may be in the last 5 bits of C(6)
without disturbing any other digits. The result may be written

V V (u & 31) ,

and may be programmed in the form

[{(u & v) ^ u} & 3l] yi V.

5.0 7 00 v to J7

6 7 05 u & V

G 1 06 (u & v) ^ u

(31) 7 45 {(u & v) ^ u} & 31

5.0 7 16

The algebra of these logical operations is called Boolean algebra; it has applications in the
logical design of digital computers. Some elementary results are quite useful; the reader may like to
consider the following relations.

U & (V V W) = (u & V) V (u & w) ,
Uw^ (V & W) = (u V v) & (u V to) ,
u & V = u s V,

U V V = U & V,

U ^ V = (u & v) V (u & v) ,
u & (v j^ w) = (u & v) T^ (u & w) ,
u & (u w v) = U V (u & v) = u.

▼

A certain operation that can be used with packed words is called extract (though this word is
sometimes used for other purposes). This operation combines three operands to give a result; it may
be defined symbolically as

(u & w) V (v & w).

- 54

THE ORDER-CODE

3.12

Here w can be thought of as a kind of mask; where the mask has 1' s the digits of the resultant word
are the same as those of u; where the mask has 0' s the result has the digits of v. For example, if

u - 0. 11110 00011 10011 00010 10101 01111 10110 110,

u = 0. 11000 11000 10101 10100 00000 10110 mil 010»

and w = 0.00000 00000 11111 11111 11111 00000 00000 000,

then the result is

0. 11000 11000 10011 00010 10101 10110 mil oio.

This operation can be used to replace one component of a packed word by a new group of digits forming a
part of another word. It can be programmed by using the identity

(« & w) V (v & w) = {(u ^ v) & w} ^ V.

To prove this we note that (u & w) & (v & w) = 0, so that

(U & W) y (V & W) = (U & W) ^ (V & 5),

= (u & w) ^ {v t- (v & w)}.

which follows from one of the and not identities. The result is therefore equal to

(a & w) ji (v & w) y^ V = {(u f^ v) & w} ^ v.

As an example of the use of this operation, let us replace digits 11 to 16 of the word (v) in 5.0 by the
corresponding digits of the word (u) in 5.1. We shall need a mask (w) having 1' s in digits 11 to 16
and O's elsewhere (i.e. O^^l'^O^S).

0.0 } mask w (pseudo order-pair)

V to X6 from 5.0

u ^ V

(u ^ v) & w in Xe

{(u -i^ v) & w} ^ V in 5.0

Note that if we add the order 5.1 6 16 we shall have interchanged digits 11 to 16 of u and the
corresponding digits of v. The mask is here written as a pseudo order-pair since this is a very
convenient way of specifying its individual binary digits; a pseudo order-pair may be defined as
something which Is written (and punched) according to the rules for order-pairs, but which is not
intended to be obeyed by the computer. We shall discuss this subject further in the next Section.

There is a useful general theorem in Boolean algebra which can be used to prove or discover
identities. We write /(u) to mean a Boolean function of u, i.e. any finite combination of u with
other letters and constants effected with the operations not, and, or, not- equivalent. The theorem
asserts that

/(u) = {u & /(-I)} ^ {u & /(O)},

and can easily be generalized to include functions of more than one variable. It is the fundamental
theorem of Boolean algebra. We can apply it to prove the extract identity above by writing

f(w) = {(a ^ v) & wf ^ V,

so that /(-I) = u ^ V ^ V = u, and /(O) = ^ v "= v. In this case the theorem shows that

77 0.

5.0

6 00

5.1

6 06

0.0

6 05

5.0

6 16

fiw)

(W & u) V (W & v) ,

which is the result of the extract operation. Note that if two functions /(u) and g(u) are to be proved
equal for all u, it is enough to prove that /(-I) = g(-l) and /(O) = g(0).

Each of the orders introduced in this Section requires only the basic time to be obeyed, i.e. 3
word- times if it is an a-order and 2 word-times if it is a 6-order.

3. 12 Orders in binary, pseudo order -pairs

It is occasionally useful to know the way in which the written form of an order-pair is represented
in binary inside the computer. This knowledge is chiefly useful when reading the monitor tubes, or when
we wish to write collating masks or certain other constants on a programme sheet for input by the
computer.

In Section 2.6 we described the way in which the 39 bits of an order-pair are used. We shall now
complete this description by an account of the way the ;V- address of an order is represented by the 7
bits used for this purpose. '

- 55

groups 0, 1, 2.

3.12 THE ORDER-CODE

If a decimal number is written in the AT-position of an order then it is directly represented in
binary. Such numbers will usually correspond to one of the following.

(a) an accumulator address

(b) the address of a special register

(d) a shift-number in a group 5 order,

(e) a main store (or buffer store) block-number in a group 7 order.

If the iV-address is the address of an ordinary register then the left-hand bit is always I; the
next three bits give the block-number in the computing store and the last three bits the position-
number. The following 'examples should make this clear.

Written form of N Binary digits representing N
0.3 1 000 Oil

2.6 1 010 110

5.2 1 101 010

In a jump-order the address of the register is followed by a + sign if the jump is to a 6-order;
in this case the A^- address is represented as above except that the left-hand bit is always 0. These
examples should clarify this.

Written form of N Binary digits representing N
0.3+ 000 Oil

2.6+ 010 110

5.2+ 101 010

The following are examples of the way the 19 bits of an order correspond with its written form.

Order as written Binary form Inside computer

N X F M

@ 5 12 0001101 101 100010 000

27 |2]72 6 0011011 010 111010 110

0.7 6 21 1000111 110 010001 000

3.2 3 66 1011010 Oil 110110 000

1.5+ 60 4 0001101 000 110000 100

When the written form of an order-pair is being converted to the binary form by the Initial Orders,
the way in which the conversion is carried out is not affected by the function parts of the orders. A
complete order-pair is assembled by shifting up the a-order and adding the 6-order and stop/ go digit.

We can, if we wish, write down "abnormal" forms of orders and get them converted to binary and
stored; the following are examples.

"Order" as written Binary form inside computer

N X F M

0.3 6 40 1000011 110 100000 000

7.5+ 2 70 3 0111101 010 111000 Oil

48 36 7 0110000 000 011110 111

Usually orders that are written like this are not intended to be obeyed by the computer. They can
however be used to express certain constants, e.g. collating masks, as pseudo order-pairs. A pseudo
order-pair may be defined as a group of symbols written on a programme sheet and punched in accordance
with the rules governing order-pairs, but which is not intended to be obeyed by the computer. A
pseudo order-pair is usually used when we wish to specify each binary digit of a constant. In Table
3.2 (page 57) are given a few examples of such constants and the way in which they can be written as
pseudo order-pairs. Note that the sign-bit (or stop/go bit) is 1 unless the pseudo order-pair is
marked as a stop order-pair. For clarity we have written dots for 0' s in the binary representation,
and written the a-order above the 6-order (this is the way words appear on the monitor tube).

THE ORDER-CODE

3.12

Collating Mask

Pseudo
Order-Pair

Abbreviated
Notation

Binary

OiOii2oi7

1 mill 111

1 77 7.
4.0 00

l4ol6il2o7

1 111

1111111 111 11

6.0 00
127 7 60

013i13q13

nil 111

mill

17 7.
7.6 00

Table 3.2 Some representative pseudo order-pairs.

It is customary to rule a heavy vertical line on the programme sheet to the left of a pseudo
order-pair, as with other constants. Sometimes collating masks can be conveniently written as numbers,
a few examples appear in the previous Section.

We have described earlier how a dummy order may be written simply as in the ^-position of the
order, the other parts being left blank. In general we can, if we wish, write some integer (unsigned)
in the ;V-position and leave the rest of the order blank; such an integer will be stored at the right-
hand end of the corresponding order. For example the last of the collating masks in Table 3.2 could
alternatively be written as follows:

When writing a single-word transfer order (function 70 or 71) a decimal main-store address can be
written in the ^V-position of the order and a minus sign in the JT-position (see Section 3.10). The
effect of such a minus sign is to cause the preceding part(s) of the order to occupy 3 extra binary
digits to the right (i.e. to be shifted down 3 places). If desired, up to four minus signs may be
written in an order, each occupying the space reserved for an octal digit in the written form of the
order (i.e. X, either of the F-digits, or M). This facility is useful primarily in pseudo order-pairs
used with modified orders and the Assembly part of the Initial Orders, which will be described later.
The following are illustrations.

Order as written

Binary form inside computer

|31 -| 71

I 3 71 71

31 - -0 1

31 - 01 -

N X F M

0000011 111 111001 000

0000000 on 111000 ooi

0000000 on 111000 ooi.

- 57

Chapter 4

Some Simple Programmes

In order to put the subjects so far discussed into their proper relationship with one another we
shall now describe some simple complete programmes. Before doing this we shall discuss briefly certain
subjects which will be more fully covered in later Chapters.

4.1 Outline of output

The output device in a basic Pegasus installation is a paper tape punch (Plate 4). When certain
orders are obeyed by the computer this punch perforates a row of holes across a blank strip of tape,
and moves the tape forward a tenth of an inch in readiness for the next row of holes. Each row of holes
is called a tape-character; there is room across the tape for up to 5 holes; each of the 32 possible
combinations of hole and no-hole is regarded as a distinct character. The tape produced by the punch
is called the output tape, in order to distinguish it from the input tape, which is the tape read by
the input tape-readers. The output tape normally goes Immediately into an interpreter, which senses
the holes produced by the output punch and prints certain characters (e. g. decimal digits or letters)
corresponding to the tape characters. When it has been printed out in this way the output tape is
often no longer needed and is thrown away. There are two main reasons for using this indirect procedure
for printing the results of a calculation:

(a) the punch is about six times as fast as the printer or interpreter,

(b) it is sometimes useful to feed the output tape back into the computer, i.e. to use it as an

input tape, on a later occasion.
To avoid circumlocution it is common to talk of the computer printing its results, whereas strictly it
only punches them for subsequent printing by an interpreter.

In order to print a number we usually have to cause the printer to print several characters; the
individual decimal digits must each be printed and we shall probably have to include a few extra
characters such as a sign (+ or -) or some spaces. Each character has to be printed as a separate
operation.

For our present purposes we can think of special register 16 (see Sec. 2. 9) as connected to the
printer. In order to print any particular character all we have to do is to send a certain small
integer to register 16. For example to print the character 5 the computer could obey the orders

3 40

16 3 10

5 to ^3

print 5

The integer we must send to register 16 to cause any particular character to be printed is called the
value of the character. The value of the character 5 is 5; the value of the character + is 10. The
characters we shall need in this chapter are tabulated, with their values, in Table 4. 1; it will be
seen that values up to 9 are allocated to the decimal digits. Some of the characters call for further

Printed character

Value

decimal point (0)

line feed (LF)

space (Sp)

multiplication sign

equals sign

carriage return (CR)

Table 4. 1 Summary of printer code

SOME SIMPLE PROGRAMMES 4.1

explanation. A decimal point Is referred to as © to prevent confusion, it is of course printed simply
as a point. The character space (abbreviated to Sp) causes the carriage of the printer to move forward
unit distance without anything being printed; its effect resembles that of the space bar on a type-
writer. The character carriage return (abbreviated to CR) causes the carriage of the printer to return
so that the next printing occurs at the extreme left of the line; line feed (LP) causes the paper to be
moved up one line so that the next printing occurs on a new line. These last two characters are usually
printed successively (as CR followed by LP, denoted by CRLP) so that the next printing occurs at the
extreme left of a new line; they could logically be combined but have to be kept separate for technical
reasons connected with the design of the printer. The characters Sp, CR, LP are sometimes collectively
called layout characters.

One should not assume that the carriage of the printer is in any particular position at the start
of the programme, and consequently most programmes start off with some such sequence as the following.

30j 6 40

] Print CR
16 6 10

13} 6 40
16 6 10
16 6 10

Print two IF' s

Note that the first two orders in the above sequence send the integer 30 to register 16 and consequently
cause the "printing" of a carriage return (CR). In order to print the integer 30 we should have to
print the characters 3 and separately, for example by means of the following orders.

Print 3

6 40
16 6 10
16 10 Print

Of the 39 bits sent to register 16 to print a character only the Is 5 bits matter, the first 34
bits of the word do not affect the character printed.

To print a number we have to calculate from the number the values of its Individual decimal
digits. To illustrate how this is done we shall construct a sequence of orders for printing a non-
negative fraction « to 3 decimal places. It is, of course, less than 1.0 and can be written Q.rst.
where r, s and t are its first 3 decimal digits. For example, it x = 0.123456... then r = 1,
s = 2, t = 3. We can easily find the value of r, the first digit, by multiplying x by 10; the
integral part of the product is just r (e.g. it x = 0.123456... then 10* = 1.23456...). After printing
r we can then replace * by the fractional part of the product and repeat the process. The following
steps should make this clear.

* =0. 123456. . .

10>p = 1.23456..., integral part = 1,
fractional part = 0. 23456. . .

10 X fractional part = 2.3456..., integral part = 2,

fractional part = 0.3456...

10 X fractional part = 3.456..., integral part = 3,

fractional part = 0.456...

After each multiplication the integral part is the next decimal digit which is to be printed. This
process requires the multiplication of a fraction by an integer; if we use a 20-order we shall get the
integral part of the product in Jr6 and the fractional part in XI (see Sec. 3.1). The following
sequence of orders can thus be used to print r, s and t it the fraction x is in 5.0:

lO; 5 40 10 to X5

5.0 5 20 lOx to ^6 and 7

16 6 10 print r

5 7 20 fractional part x 10

16 6 10 print s

5 7 20 fractional part x 10

16 6 10 print t

It would be better to print a zero and a decimal point first and rearrange the above orders into
a small loop as follows.

- 59

4.1

SOME SIMPLE PROGRAMMES

16 10
@ 5 40

0.0

Q) 4 40

(l2) 6 40

5.0 7 00

ft.

16 6 10

5 7 20

0.2+ 4 67

print zero

10 to X5

set counter = 4 in J4

12 = code for decimal point, to ^"6

X to XI

Print decimal point or digit

fractional part (or x) x 10

count down from 4 (© and 3 digits)

In practice other points would be taken into account. For example, we should print either CRLP or
SpSp before the number; and we would probably arrange to print the sign so as to allow x to be
negative (if x < we must change its sign before entering the above loop of orders). We should also
take care to round the value of x before printing (by adding 0.0005 to it) since we are only printing
the first few digits of a long number (c. f. the use of rounding in multiplication, division and
shifting). We must then consider the possibility of overflow caused by the rounding or the change of
sign. When all these points (and others) have been taken into account we get a self-contained piece
of programme of moderate complexity. Such a piece of programme is called a subroutine; a routine is
simply another name for a programme, and a subroutine is a fairly independent, self-contained part of a
routine. We shall discuss this subject further in the next Section.

Let us now consider how integers may be printed. To start with we shall consider the printing of
a non-negative integer a less than 100. One way is simply to divide the integer by 100 to convert it
into a fraction and then to use a cycle of orders similar to the one given above; this method is
awkward for larger integers so we shall not consider it further. If r and s denote the two digits of a,
then a = lOr + s, and we can find r and s by dividing a by 10. For example, the following orders
could be used, supposing a to be in 5.0 at the outset.

0.0

5.0

7 00

6 40

7 10

6 10

a to Xl

10 to xe

divide a by 10
print r
print s

This sequence would probably not be used in practice since it does not incorporate any suppression of
left-hand zeros, i.e. an integer such as 7 would be printed as 07. We must arrange that the first
digit is not printed if it is zero. The following sequence shows how this can be done.

— ►

5.0

7 00

0.0

6 40

0.2+

7 60

7 10

L — ^

6 10

* divide a by 10

omit first digit if zero
print first digit if not zero
print second digit

This device is known as left-hand zero omission. Even this sequence would not be acceptable in most
programmes, since the width of the printing is not constant, i.e. sometimes the printing will be two
units wide and sometimes only one. This is disadvantageous when several numbers are printed on the
same line and similar lines are printed repeatedly: the layout of the printed page is then irregular.
To obviate this we must arrange to print a space instead of the first digit if it is zero; this is
called zero suppression.

60 -

SOME SIMPLE PROGRAMMES

4.1

— ►

5.0

0.0

0.2+

^ f*

divide a by 10

if first digit is zero replace it
by 14 (code for space)

Print first digit or space (=14)
Print second digit

Generally speaking we should. want to print larger integers than can be handled by the above
sequence of orders. To see how this can be done consider the printing of the integer a = 123456. If
we divide it by 10 the remainder will be 6 and the quotient 12345; if this quotient is then divided by
10 the remainder will be 5 and the quotient 1234. This process can be repeated, but has the disadvantage
of producing the digits (as the remainders) in the reverse order. A better way is to divide o by 100000
to give a quotient of 1 and a remainder of 23456. The quotient can then be printed and the remainder
multiplied by 10 to give 234560. This number is in turn divided by 100000 to give a quotient of 2 and a
remainder of 34560. The quotient is printed and the remainder multiplied by 10 again, and so on. The
following table shows how the successive quotients give the required digits.

quotient

remainder

remainder x 10

123456 -r 100000

23456

234560

234560 -^ 100000

34560

345600

345600^ 100000

45600

456000

456000 -r 100000

56000

560000

560000 -r 100000

60000

600000

600000 4- 100000

The following orders will carry this process out for any 6-digit non- negative integer.

0.0

+100000

G
@

4 40

5 40

w
1

5.0
0.0

7 00
24

7 10
6 20

0.2+

4 67

Set counter for 6 digits

10 to Xb

a to Xl

divide by 100000

print digit

remainder x 10 to XI

count digits

As" before, we must arrange for left-hand zero omission or suppression, the details of which are left to
the reader. In Section 4.4 below we give an example of a sequence for printing single-length positive
integers of any magnitude. As when printing fractions there are a number of complications and in
general the whole process is best left to a subroutine.

As a rule the printing produced by a programme will consist of a great many numbers, some of these
will be printed as integers, some as fractions, some as mixed numbers, some in sterling, some will be
signed, etc. It is well to arrange that the printing is neatly laid out in columns, blocks, and so on,
and this can be arranged by suitable counting and printing of layout symbols. It should be noted that
there is room for printing up to 69 characters on one line of the printer.

In this section we have described the way in which numbers can be printed using the equipment in a
basic Pegasus installation. One can also easily arrange to print alphabetical and some other
characters, and we describe this in Section 6.2. The normal output punch operates at'60t characters

t Many Installations will still be using the slower output punch which operates at 33 characters per
second.

61 -

4.2

SOME SIMPLE PROGRAMMES

per second and this is adequate for most scientific and technical calculations. Sometimes, however,
greater speed is necessary and the basic installation must then be supplemented by some ancillary
equipment: this is most needed in large-scale commercial and statistical work. Some of the extra input
and output facilities are described in Chapters 10 and II.

4.2 Subroutines and the organisation of a programme

A reader new to the subject could be forgiven if, at this stage, he felt that the task of preparing
a complete programme was altogether too difficult to be contemplated. It is true that the preparation
of a large and complicated programme is an operation which may take a year or more; but there are many
ways of lightening and spreading the labour, and much of it is, by any standards, interesting. Nearly
all programmes can be subdivided into self-contained parts, each doing some job which can be regarded as
more or less complete and independent: these parts are often called subroutines. If a flow-diagram of
the whole programme is available then each box, or some convenient group of boxes, can be thought of as
a subroutine. The Importance of the idea is that, provided a sufficiently precise specification of its
function can be written, each subroutine can be programmed and tested separately, without much reference
to the rest of the programme. In this way the programme can be tackled piecemeal, if necessary by
several people. Afterwards the various subroutines can be linked together to make up the complete
programme.

This procedure is of the greatest value in the preparation of a complicated programme. We can
visualize a programme organized in this way as a collection of subroutines so arranged that when one
has finished its task it leads into (or calls in) the next. As a rule there will be a few "red tape"
or organisational orders to be obeyed between one subroutine and the next, especially if they have been
written by different people, and it is therefore often more convenient to have, in addition to the
subroutines, a master-programme, (or master- routine) which includes most of the organisational orders
and whose main purpose is to call in the various subroutines in the right sequence. If the programme is
organised in this way, which is usually the case, the subroutines must be so written that they re-enter
(or return control to) the master-programme when they have finished their work. In addition to calling
in the various subroutines, the master-programme usually performs a few other operations, such as counting,
moving numbers from one part of the store to another or doing a small amount of arithmetic. It often
happens that some of the subroutines may themselves call in other subroutines for some subsidiary calcu-
lation (these are sometimes referred to as sub-subroutines) so that the whole structure of the programme
may be on several levels as illustrated in the diagram (Pig. 4. l).t

MASTER

- P

R A M M E

Subroutine

Subroutine D

(used by

subroutine B)

Pig. 4.1 Structure of a typical programme

It will be noticed that subroutine A has been called in (or entered) twice from two different
points in the master- progrMmie.

The orders used to call in a subroutine are called the cue to the subroutine. A cue is normally
an order-pair consisting of a block-read order and an unconditional jump. Por example, the following
is a typical cue:

B37 to f/0
jump to 0. 1

When such a cue is obeyed a block of the subroutine (usually, but not always, the first block) is
copied into the computing store and entered (i.e. a jump to one of its orders occurs). After this the
computer is obeying the orders of the subroutine, which will include any further block-read orders
needed. Eventually the subroutine will have finished its task (e.g. printing a number, evaluating a
square-root, reading in a list of numbers, etc.), and it then obeys a link, which is the order used to
leave (or exit from) the subroutine, usually so as to return to the master-programme. The link is
often an order-pair made up, like the cue, of a block-read order and a jump.

As an illustration, suppose that there is a subroutine for printing (7(7) as an integer which is
to be entered by the cue written above. At the relevant point in the master-programme we might have
the following orders.

t In this diagram the box representing the master-programme has, of necessity, been made larger than
those representing the subroutines. This Is not intended to reflect the amount of storage space
occupied.

- 62 -

SOME SIMPLE PROGRAMMES

4.2

START-

0.0

A4-
B —

B 10

5.0 7 00
(J) 5 40

37 El 72
0. 1 60

5.0 7 00
7 7 20

• cue to subroutine

In the subroutine there could be the following orders.

B 37

0,0

+1000000

(Jo) 4 40

A ■

38 IE 72

0.0 (

) 24

B38 to Ul

1.5

etc. . .

10 10]72

0.2 60

> link

When the link is obeyed in the subroutine the original block (BIO) of the master-programme is put back
in f/0 and re-entered at the a-order in 0.2 (marked B). A subroutine written like this can, of course,
be entered from several different places in the master-programme but, since it always obeys the same
link, it will on every occasion return to the same point in the master-programme. Frequently this
would not be what was wanted and in a programme like that of Pig. 4.1 would necessitate the storing of
two (or more) copies of a subroutine, each with its own fixed link. It is better to arrange that the
link obeyed by the subroutine can be changed so that return to the master-programme is to different
points on different occasions.

The usual arrangement is as follows. The master-programme places the appropriate link in
accumulator 1 just before obeying the cue which calls in the subroutine; this process is called setting
the link. The subroutine is written in such a way that, when it has finished its task, it copies the
link into an ordinary register (this is called planting the link) and obeys it. For example, the
master-programme might contain the following sequence of orders.

B 10

5.0 '

r 00

0.0

0.2 ]

L 00

37 G

372

A .rf

0.1 (

} 60

a 72

0.3 (

3 60

5.0

7 00

1*^
3

7 20

set link from 0.2 in XI

' cue to subroutine

« link for subroutine

And in the subroutine there could be the following orders.

- 63 -

4.2

SOME SIMPLE PROGRAMMES

B 37

0.0

+1000000

(lo) 4 40

38 [r|72

0.0

1.5

etc..

•

7 1

B38 to Ul

plant link from XI in 0.7
obey link

It will be noticed that the link is written in the master-programme (in 0.2) immediately after the cue;
this is often a convenient place to write it. The link is marked with a heavy line on the left like a
constant or a pseudo order-pair; this is to draw attention to the fact that it is not obeyed in the
master-programme, where it is written, but in the subroutine (actually in UO.l). With this device the
subroutine can be entered from several different points in the master-programme; on each occasion the
master-programme can set a different link in XI. Of course, there is no need to use XI for the link
in this way, another accumulator (or an ordinary register) could be used but it is conventional to useA^l.

It is essential that the subroutine should carefully preserve the link until it is needed. If the
subroutine requires to use XI then it should copy the link elsewhere first; this may happen because the
subroutine requires a single-word transfer order, or perhaps because it calls in a further subroutine.

When a subroutine has been written it is advisable to prepare a specification of it. This is simply
a catalogue of the properties and effects of the subroutine as viewed from outside. The specification
would state, for example, which blocks in the computing store were used by the subroutine, what cue
(or cues) are to be used to enter it, which accumulators it uses as working space, what its precise
effects are, and so on. There are a number of conventions relating to subroutines, among which the
following are important.

(a) the link is to be in XI on entry,

(b) OVR is to be clear on entry,

(c) nothing should be assumed about the contents of any registers or storage locations used as
working space by the subroutine,

(d) if possible, the subroutine should use blocks I/O and Ul only in the computing store, and also as
few accumulators as possible.

All these things are purely conventional. It is important to distinguish between the rules of
programming, which are necessary because of the way the computer works, and conventions, which form a
body of accepted practice. An adequate set of programming conventions is an essential prerequisite
for the satisfactory use of a computer.

It is usually necessary to enter a subroutine by a proper cue as described above (i.e. a block-read
and Jump) even if the subroutine is used several times in quick succession. This is because most sub-
routines write over some of their own orders when these are no longer needed; consequently a fresh
copy of the routine must be brought in from the main store on each occasion when it is used. Since some
subroutines use ^1 as working space, the link must be set each time the subroutine is used, even if the
same link is required again.

The following are some imaginary, but plausible, brief specifications of subroutines.

(1) Print three-digit non- negative integer in XT, preceded by SpSp.

Cue:

40 E] 72
0.0 60

(2)

Uses: UO, BO.

Link: In ^1, obeyed in 0.3.

Time: about 100 milliseconds.

Evaluate y(pq)r> and leave result in XG.

Cue:

43 [0]72
0.0 60

- 64 -

SOME SIMPLE PROGRAMMES

4.2

(3)

Uses: UO; X5. 6, 7.

Link: in XI, obeyed in 0.7.

Time: about 45 milliseconds.

Read a sum of money in sterling from the input tape, convert it to pence and leave the
result in X6.

Cue:

44 [g]72
0. I 60

Uses: UO; J4, 5, 6, 7.
Link: in XI. obeyed in 0.7.
Time: about 108 milliseconds.

Some subroutines may usefully be written so as to have two or more alternative entry-points, each
with its own cue. For example, a print subroutine could precede the main printing by CRLP if entered
by the cue

40 072
0.0 60

or by SpSp if entered by the cue

40 [0j72
0.2+ 60

Sometimes it is advantageous to use a device known as a programme-parameter. This is usually a
number set in an accumulator by the master-programme just before calling in a subroutine. The effects
of the subroutine are then determined to some extent by the parameter. A straightforward example is
provided by a subroutine for printing a fraction; the subroutine could be so written that the number
of decimal places printed is determined by, say, an integer in X2, The number of places required is
then set in X2 by the master-programme at the time it sets the link in A^l, i.e. just before obeying the
cue to the subroutine. Like the link (which is really only a special kind of programme-parameter), the
number of decimals wanted is set each time the subroutine is called in and can easily be changed.

We have already pointed out that the use of subroutines can greatly simplify the task of preparing
a complete programme by enabling the whole job to be broken down into parts which can be dealt with more
or less independently. There is a further advantage: some subroutines may be used in more than one
programme. For example, many programmes will need a subroutine for printing an integer. Such a
subroutine can be written by an experienced programmer in such a way as to be efficient and to have
features making it widely useful. Many generally useful subroutines like this have been written, they
together form the Library of the computer and there are volumes available which contain their
specifications. The specification of a library subroutine provides all the information needed for the
normal use of the subroutine (e.g. details of cues, working space, and so on); it does not indicate,
nor need the user know, how the subroutine works. Since the use of the library requires a knowledge of
the Assembly section of the Initial Orders, we defer further discussion until Chapter 8.

In most scientific and technical calculations functions of various kinds are needed. For ex^ple,
square-roots, logarithms, exponentials, sines, cosines, and so on, enter into many calculations. When
carrying out a computation by hand it is usual to use books of tables for such functions. With a
digital computer one could, of course, store a table, together with a subroutine for looking up entries
in the table and interpolating in it. This is sometimes a useful technique but it should be realized
that such a table may occupy a considerable amount of storage space; furthermore the interpolation
routine is by no means trivial and may be relatively slow. It is usually best to evaluate functions as
they are needed; for example, sin x may be evaluated by means of a truncated Taylor series

sin% ~ * -

3! 5!

+ (-)"

^2n + l
(2n+l)!

Only six or even seven terms are needed to give the full precision attainable with 39-bit words, and one
can therefore construct a fast subroutine to evaluate sin* (the numerical values of the coefficients
being stored within the subroutine). (This is, in fact, not the best way to evaluate sinaj. )

As a simple illustration of how subroutines can be used let us prepare a table of powers of 2.
We shall print two columns giving the values of n and 2", respectively, starting at n = 1 and going on
until 2" is no longer a single- length number (the last value will be 2^''). We shall assume that a

- 65 -

*-2 SOME SIMPLE PROGRAMMES

subroutine is available with the following brief specificatlont:
(4) Print integer In JH (with zero suppression).

Cues:

(1)

50 [1]72
0.0 60

to precede integer by CR LP

or (2)

50 [0)72
0.3 60

to precede Integer by Sp

Uses: UO, BO.

Link: in A'l, obeyed in 0.3.

Programme-parameter: integer in X2 specifies maximum number of decimal digits
in number being printed.

We know that n will have at most 2 digits and 2" at most 12, so we can set the programme- parameter to
these values. We shall keep the current value of n in X5 and that of 2" in 13. The master- programme
is a little under two blocks long; we shall suppose it occupies B2 and 83 and can be entered by
copying these two blocks into UO and Ul and jumping to 0.0; we shall describe how this is done, with
the help of the Initial Orders, in the next Section.

B 2

START ►

(li)

6 40

0.0

6 10

5 40

3 40

7 00

n ^

2 40

0.5

1 00

[S372

0.0

/■ V '

U bv

[0]72

1 5

0.6

\. ih

7 00

2 40

1.1

1 00

^ print LP LP (CR unnecessary here
because it is done by the subroutine)

set n = 1 in X5 1
set 2" = 2 in JfS J
n to J!r7
parameter = 2
set link 1 in XI

'Cue (1) to subroutine

■link 1

2" to XI
parameter = 12
set link 2 in XI
dummy order

initial values

print n on a
• new line (maximum
of 2 digits)

print 2"
on same line
(maximum of
12 digits)

It will be noticed that this print subroutine uses only UO In the computing store. This is typical
of output subroutines in general, which are usually written so as to preserve all the accumulators
by writing them into BO on entry (there is an order 73 at the beginning of the subroutine)-
the accumulators are reset from BO just before the link is obeyed. The time required by these extri
block-transfers (and others in the subroutine) is generally small compared with the time needed to
operate the output punch.

SOME SIMPLE PROGRAMMES

4.3

B 3

1.0

50 072
0,3 60

1 1

2 [0]72
1.2 60

'. B^

n 9+ n c^

1.3+

cue (2) to
subroutine

► link 2

print 2"
on same line
" (maximum of
12 digits)

increase n by 1 in X5
double 2" in X3
repeat if OVR clear
stop if OVR set

Note the dummy order (in 0.7+) inserted to make the second cue to the subroutine an order-pair (although
this is strictly not necessary here). On each occasion the link restores B2 in UO and this is all we
need to do after using the subroutine since it uses only UO in the computing store.

4.3 Putting a prograiwne into the computer

We have so far described the effects of various orders and how they can be written, together with
some numbers, on programme sheets. We have also discussed the way in which programmes can be organised
with the aid of subroutines, and how the results are printed. Let us now consider how a complete
programme and the numerical data for it can be supplied to the computer. We shall suppose that the
complete programme (including all the subroutines) is available and that we wish to place it all in
specified places in the main store, after which the beginning of the master-programme is to be copied
into the computing store and the machine is to start obeying its orders at some specified poirtt.

We must first of all prepare the programme tape. This is usually done with a teleprinter forming
part of the tape-editing equipment (see Plates 7.10). This can be thought of as a kind of typewriter
on which the orders and numbers of the programme can be typed out; as a result of this the teleprinter
produces a printed sheet and a length of punched paper tape. With few exceptions, each time one of
the keys on the teleprinter keyboard is pressed a single character is printed and the corresponding
tape-character (i.e. a certain row of holes) is punched. We shall not at present describe the rules for
operating the teleprinter and punching the tape; they are such that the way in which the orders and
numbers appear on the printed sheet differs only slightly from the way they are written on the
original programme sheets. The punching rules are consequently natural, and easy to learn and apply.

The printed sheet (known as a print-oaf) can be proof-read against the programme sheet, and this
is usually an adequate way of checking the punching of the programme. Pig. 4. 3 shows a programme sheet,
and Pig. 4. 4 the corresponding print-out. The data on which the programme is to operate are pmiched in
much the same way but must be much more carefully checked; it is usual to punch the data twice and to
compare the resulting tapes automatically in the comparator section of the tape-editing equipment.
Should the checking of the tapes reveal any errors in the punching, these can easily be corrected in
various ways with the help of the tape-editing equipment. If the programme is a long one it may be
punched in sections (for example, each subroutine could be punched separately) and the several lengths
of paper tape Joined together (or spliced) afterwards. When the programme tape is complete it is
spooled up on a hand spooler (Plate 6) and placed in a box (Plate 5).

When the programme is to be run on the computer the programme tape is usually put into a tape
trough and its start is placed in the main tape- reader on the computer desk (Plate 3). There is always
a leader of blank tape at the beginning of the tape and it is this which is placed in the tape-reader.
The Initial Orders are then called in by operating the Start key on the control panel.

The Initial Orders are the principal means by which programmes and numbers are fed into the
computer; they form a complicated programme stored permanently in the isolated part of the main store.
They are complicated because they are both easy to use and comprehensive in function. To call them in
we must first put the Run key to the STOP position and then push the Start key up to the position
labelled START (this key springs back to the NORMAL position when released). This operation clears
OVR and the external-conditioning relays (see Sec. 3. 10) and certain stops (write-with-overflow, and
unassigned order, see Sec. 3.9), prepares the computer to obey an a-order next, and places the start
order-pair, which is

896 072

0.0 60

in the order- register. Now fi896 is the first block in the isolated part of the main store; it
contains the beginning of the Initial Orders programme, so that when the Run key is put back to RUN
and the start order-pair is obeyed it causes the computer to start obeying the Initial Orders, t
Assuming the handswitches to be clear, which is usually so, the Initial Orders then read the tape in
the main tape-reader, taking in the various orders and numbers punched in it, converting them to the

' For the 4096-word store, the start order-pair is

512 ©72
0.0 60

- 67

4.3

SOME SIMPLE PROGRAMMES

internal form required by the computer and placing them In the main store. The process of calling in
the Initial Orders in the way Just described is called the Normal Start.

We must be able to direct the operations of the Initial Orders in at least two important ways:

(a) we must control where, in the main store, the orders and numbers read from
the tape are placed,
and (b) we must arrange that, on reaching the end of the tape, the computer starts to obey
(i.e. enters) the programme just read in.

These and other operations of the Initial Orders are controlled by special groups of characters,
called directives or warning characters, punched along with the programme in the tape. The directives
are used exclusively to direct the operations of the Initial Orders, they are not stored. Before
discussing them further we must introduce the Transfer Address.

The Transfer Address (usually abbreviated to T. A. ) is simply the main store address where the next
order-pair or number on the tape is about to be placed by the Initial Orders. Every time an order-pair
or number has been read in and converted into a binary word, this word is written into the storage
location specified in the Transfer Address: as soon as this has been done the Transfer Address is
increased by unity in readiness for the next word. Consequently the order-pairs and numbers punched
in the tape are placed in consecutive storage locations in the order in which they are punched. We can
easily set the Transfer Address to any value we, please at the beginning of the tape, or in the middle
of it, by using the warning character T (for transfer). Suppose, for example, that we wish to place the
first order-pair on the tape in B51.2; all we do is to punch T 51.2 at the beginning of the tape
before the first order. The T 51.2 is a directive (in which T Is the warning character and 51.2 the
address) and is not stored; it merely causes the Initial Orders to set the Transfer Address equal to
51.2 and then carry on to read more tape. If there is a subroutine on the tape which has to be stored
in B19.0 onwards, then we can punch the directive T 19.0 in the tape just before the subroutine.

The Transfer Address is set equal to 2.0 during a Normal Start; consequently if there is no
T-directive at the head of the tape, the first order-pair or number read will be placed in B2.0,

When the Initial Orders reach the end of the tape we usually want the computer to enter the
programme (i.e. to start obeying it); this can be done by using another directive consisting of the
letter E (for enter) followed by an address. The effects of this are best explained by an example:
consider, therefore, the result of the directive E 51. 2, which means enter (start obeying orders) at
the a-order in B51.2. As soon as this has been read by the Initial Orders (probably at the end of
the programme tape) there is a 77-stop; this is to allow the input tape to be changed or the hand-
switches to be set, etc. When the Run key is operated, B51 (the block containing the specified order)
is copied into UO, and the next three blocks (B52, 53 and 54) are copied into Ul. 2 and 3,
respectively. The Initial Orders thus transfer the first four blocks of the programme into the
computing store. After this there is a jump to the specified order, now the o-order in U0.2. and
the computer leaves the Initial Orders and starts to obey the programme. There are a few other effects
of less importance which we shall describe later, but it will be seen that when the Initial Orders
read E 51.2 from the input tape the effect is similar to that of the following sequence of orders.

77
51 072

4.0

4.1

52 [T]72

53 gl72

4.2

54 |3]72
0.2 60

stop, wait for Run key.

transfer four consecutive
► blocks of progranmie into
the computing store.

enter the programme.

If we want to enter the programme at the 6-order in 51,2, instead of the a-order, all we need do is to
use E 51. 2+, i.e. we write a plus sign after the address (like we do with jump orders).

Usually the directives associated with a programme are written in the appropriate places on the
programme sheets; but it is important to realize that they do not form part of the programme when it
is obeyed. The directives are concerned only with the process of input of the programme and not with
its execution.

As an illustration of the use of these directives, suppose we have a programme consisting of three
parts:

(a) a master-programme which is to go into blocks J?5 to 24,

(b) a list of numbers which is to go into i?233. 5 onwards, and

Suppose the programme starts at the fc-order in 5. 2. The tape could be punched as follows:

T 5.0

(master-programme)

T 233.5

(list of numbers)

T 50.0

(subroutine)

E 5.2+

- 68

SOME SIMPLE PROGRAMMES 4.3

The address In a directive always refers to the main store. The address in a T-dlrective can be
the address of any location in the main store. That in an E-directive can be the address of any order
in the main store, but this order must be obeyed in f/0. There are many other directives which will be
described in detail in later chapters, but we shall suimiarize three of them briefly here.

The J-directive (J for jump) is very similar to the E-dlrective; the only difference being that
there is no 77-stop before the programme is entered. Thus the directive J 51.2+ punched in the tape
causes a programme to be entered at the fc-order in B51.2 as soon as it has been read.

In order to prevent wrong tapes being used every tape should have a name. This name should be
written in ink or pencil on the blank leader at the beginning of the tape. As a valuable extra pre-
caution the name is usually also punched before the first order or number in the tape; it is then
preceded by N (for name). The warning character N, followed by the name and a short length of blank
tape, form an N-directive (there is no address). When this is read by the Initial Orders the whole
name is copied from the input tape on to the output tape. i.e. the name is printed; the end of the name
is signalled by the short length of blank tape. The name itself can be made up of any of the available
characters but must not, for obvious reasons, include any lengths of blank tape. It is advisable to have
names for each part of a complete programme, e.g. for each subroutine; normally each name would start
with CRLF so that it gets printed during input on a line to itself.

It is desirable that the output of the programme should bear the date, and to this end each complete
programme tape should be headed by a D-directive, punched immediately after the blank leader before
anything else in the tape. This consists simply of a letter D; there is no address. When the Initial
Orders read a D-directive they cause the date and a serial number to be printed; the serial number is
increased by one every time it is used. In this way the printed page produced by the programme is
headed automatically by the date and serial number on each occasion the programme is used. It is then
easy, on a later occasion, to determine the dates when various runs of a programme were made. This is
especially useful when a new programme is being developed; the effects of various alterations made to
the programme between successive runs can be studied without risk of confusing the runs. t

The directives so far described are summarized below; here the address is written as a if it is
the address of a storage location, or as a(+) if it is the address of a single order.

T a set Transfer Address = a

E a(+) enter programme at a(+) after 77-stop

J a(+) enter programme at a(+)

N print the following name

D print the date and serial number

As an illustration, suppose a programme is made up as follows:

(a) the master- pro gramme is to occupy B2 to B12.

(b) a subroutine to go into 518 onwards has the name SPECIAL PRINT 3,

If the name of the programme is DTRY 73 and it is to be entered in B2T0 then its tape could be
punched as follows.

DTRY 73

T 2.0

(master-programme)

T 18.0

SPECIAL PRINT 3

(subroutine)

T 54.3

CONST. PZ

(numbers)

E 2.0

When this tape is read by the Initial Orders there is printing like thistt before the programme starts
to be obeyed.

t The date and serial number (initially zero) are set in B895.1 and B895.6 respectively as a part
of the daily maintenance process. These are the last two storage locations before the Isolated
part of the main store, and programmers should avoid writing into them. In the 4096-word store,
B511,7 and B511.6 are used.

+t We assume here that HO = 1, i.e. that the sign-digit key of the handswitches is down. If ffO =
then some extra printing occurs when certain directives are read; this is called optional printing
and will be described later.

- 69 -

4.4 SOME SIMPLE PROGRAMMES

14/8/60- — 89
DTRY 73

SPECIAL FRIOT 3
CONST. PZ

Here the top line consists of the date and serial number. The directive T 2. punched befoi*e the
master-programme can be omitted because the Transfer Address is always set to 2.0 at the beginning by
the Normal Start operation.

4.4 A simple complete progranmie - "Special FactoriiKe"

We shall now describe a simple complete programme for factorizing whole numbers. For example, if
supplied with the number 420 it will cause the computer to print.

420 = 2x2x3x5x7

The process used may be described by the flow-diagram of Pig. 4.2,- in which the following notation
is used.

N At first this is the number to be factorized, but it is reduced by having factors removed as
they are found,

d Trial factor, initially 3, then successive odd numbers,

q The quotient when N is divided by d,

r The remainder when N is divided by d.

This process falls naturally into three stages, as follows.

Stage A The number (N) to be factorized is printed on a new line, and followed by an equals sign.

Stage B The number is repeatedly divided by 2 as many times as possible (if any), and '2' is printed
for each factor. A multiplication sign is also printed after each factor if there are any more factors,
i.e. if the quotient is not 1. As each factor is found it is removed from N, i.e. N is replaced by the
quotient.

Stage C The number (N), which is by now odd, is divided by a trial factor (denoted by d). This trial
factor is at first 3, and is then increased by 2 repeatedly so that we try dividing N by successive
odd numbers 3, 5, 7, 9, ... If one of these trial divisions leaves a remainder of zero then of is a
factor of N and it is printed and removed from N (i.e. A^ is replaced by the quotient). A multiplication
sign is also printed after each factor if the quotient is not 1. After each unsuccessful trial
division d is compared with the quotient g; if d is less than q we try the next value of d, but
otherwise N is prime and is in turn printed (see the discussion in example (c) of Sec. 1.5).

To get a natural layout in the printing we must put a space on either side of the equals and
multiplication signs, and we must arrange to omit left-hand zeros in the numbers (see Sec. 4.1). To
show how integers may be printed with left-hand zero omission we have incorporated the necessary
sequence of orders in the programme instead of using a proper subroutine of the kind described in
Sec. 4. 2. This particular sequence is especially simple because the integers to be printed are known
to be positive (no special treatment of the last digit is needed) and it Is called in at only two
points in the programme, firstly to print N and later to print the factors. Instead of using a link,
XI is used as an indicator to steer the computer to the right part of the programme: XI is clear when the
the sequence is printing Z^, but thereafter XI holds 14 (the value of Sp).

The whole programme fits into four blocks which are placed in B2 to 35 in the main store and are
copied into UO to f/3 when the E-directive is read (see Sec. 4.3). Consequently the programme proper
is concerned exclusively with the computing storie. The number to be factorized is placed in B2.0
along with the programme and is therefore available in UO.Q on entry; this number is initially 420
but can be changed as we shall shortly describe. The programme is entered when the Initial Orders read
E 2. 1 or J 2. 1. A facsimile of the programme sheet is shown in Pig. 4.3.

The three orders starting at the 6-order in U3. 5 at the end of the programme require some explanation.
A jump to these orders occurs if N is negative or when N has been completely factorized. Since U3.6
holds a stop order-pair (see Sec. 3. 9) the first effect is to cause an optional stop. When the Run key is
operated to allow the computer to continue, the effect of these orders is to call in the Initial Orders,
exactly as though a Normal Start operation had been done. Why this happens will be described in a
later chapter since a knowledge of modification is needed. The reader will recall that in a Normal
Start the Transfer Address is set to 2.0, after which the Initial Orders read tape. If therefore we now
put in the tape-reader a tape such as the following

+ 996
J 2.1

then the Initial Orders will put the integer +996 (or whatever other number is on the tape) into
B2.0 (overwriting the previous number there) and then enter the factorizing programme as soon as the
J 2. 1 is read. We can therefore factorize several numbers in succession by simply preparing an input
tape on which each number is followed by J 2. 1. If this is done then it is best to inhibit the optional
stops by means of the special key on the control panel. This input tape should be placed in the main
tape-reader when the programme tape has all been read in by the Initial Orders. Note that we are
in effect using the Initial Orders as a kind of subroutine to read in the numbers; this subject is
further discussed in Chapter 7.

- 70

SOME SIMPLE PROGRAMMES

4.4

START

Print "Carriage Return" and two "Line- Feeds'
Test sign of A^ ► STOP

negative

positive
(or zero)

Print the value of N followed by "="

B Divide N by 2 (quotient = q, remainder = r)

Is r zero?

YES (N is even)

-► Replace N by q
Print "2"

Is g = 1?

YES

STOP

multiplication

sign

▲

▼
Set rf = 1

;

Increase rf by 2

N.B. d is the trial divisor

Divide N by d (quotient = q, remainder = r)

Is r zero?

YES (i.e. d is a. factor of N)

(i.e. d is not

a factor of N)

Evaluate d - q
Test sign ot d • q

Positive or zero
{N is prime)

Negative

Try next
value of d

Replace d by N
Set q = 1

Replace N by q
Print value of d

multiplication

sign

Pig. 4, 2 Flow- diagram of the programme 'Special Factorize*

4.4

SOME SIMPLE PROGRAMMES

ruuNTi
Date 18. 5*. 57

PEGASUS PROGRAMME SHEET

Sheet

D
N
SPECIAL FACTOKIZE

0.0

[star

/ ♦

+420

4.0

Vg/ ' "*"

1(,

It
0. o

d.sr-t-

3.7

1.1

3
0.7

3.1 +

It
It

<7.0

5
2.3+

O.O

40
10

4o
40

00
00

24
01

2,0

20
t7

t1
40

10
40

40
OO

Z4
t1

io
io

N s 4-20 J ^nu/mbcr
t» be ■fa.cfirrtxc^ .

ISctU

Line Fceolj

5 TOP If i»e««itfc'i«f

3-<

-i« 3

'10

Omit

frhil;

fm.%i

in X7

It
It

1.5-+

0./>

3.0

2.4

O.O

*s4
3.0

s,. ,

Print S^tLCe /i

JiMMj, \C I- t5 not z«n» (iv «(m)
Rc^Um N fry ^ fTNO-tfjll/
PoHi: "2" 1^

0.5"

O.O

3.5+

It
It

2.4 +

O
o. o

43
GO

10
10

41
oo

04
t3

B5-

(?0

43
to

72
to

STOP if

f^etUm ^r next -f<«.ct"0r

I remaiMiJler = r t» Xfe.
JJum^ i^' r is zero; i.e.
"1^ if «A iix A fachtr of W,

Fv>rtM d-a -u* X7
fJumt^ i^ Mea«,ti««y «-«. if

Set«^=1 i« X7 juse

5eti.Nl«X5jp;rr

Ti-f lOOOOooooqoo

(Cofy d. ith. X7
I 1^ t'^Hfc^uj.

STOP

Q>fy N ink, X7

W~1 ...X7
TVimJ" Sna.ce I RefcUrM ^

4S 1

I next

"1 fKwt p<

I liuiU*yUcnt£oM L J

J ^•^•^ I »

I -iy net: 1.

} 5top ( ofytionol ) , tUen
CA.U. CH Imti«i| Ordus

as 1C>" («*>€il fwr

«A^ ««. »ve«ii

E 2.1

Pig. 4. 3 The programme sheet for 'Special Factorize'.

SOME SIMPLE PROGRAMMES

4.4

Directives: -
D (Date & serial No. )

«N (Name)

These cause printing
when the programme
tape is read into
the computer.

SPECIAL FACTOR I ZE

+ 430
30 640

13
16
16
o*o

6zo
640
6io
610
700

3,5+763
too
240

3 4«>
400
600
634
401

12
10
O
O

3»
7

Read into block
B2 of main store

i»x 460
16 710
3 630
0.7 367
3« 1+161

14
16
36
16
16
3

X 40
1 10
640
6io
110

540
700

034

3«3+66i
o#o 710
16 510

Read into block
B3 of main store

I 7 43
3.5+760
16 no
34 640
16 610
16 no
!• 5+060

I
2
o*o

5
3.0

2*4

z
o*o

540
541

700
034

660
704
763
740
500

Read into block
B4 of main store

Directive E 2. 1

(causes stop, after
which B2 of main store
is copied into block
of computing store;
B3, B4 & B5 are
similarly transferred
to blocks 1, 2 & 3
resp. in computing
store; then jump
to 0.1).

o»o 710

5 700
0.5 060
o.o 700

I 743
3.5+760
16 no
34 640
i6 610
16 1 10
3. 4+060

37 400
o 0724.
o.o 060
+100000000000

^E 3.1

Read into block
B5 of main store

These stars mark tail
end of tape

Pig. 4.4 Print'out of the programme tape for 'Special Factorize'.

' 73

4.5 SOME SIMPLE PROGRAMMES

Pig. 4.4 is a reproduction of the print-out obtained when the programme is punched; this should
be compared with the programme sheet of Pig. 4. 3. Note the Stars punched at the end of the tape -
these are used to distinguish one end of the tape from the other. The print-out shown in Pig. 4.5 is
of a typical input tape carrying various numbers to be factorized: in this print-out each J 2. 1
appears on the same line as the number; this is permissible, as will be explained later, since a
number may be terminated either by CRLP or by Sp when it is to be read by the Initial Orders. After
the last number on this tape is punched a Z-directive; the effect of this is simply to cause the
Initial Orders to encounter a 77-stop; it is punched here just to stop the computer when all the
numbers have been factorized. The results of running this programme are shown in Pig. 4.6, which is
a reproduction of the output.

TEST NUMBERS

+4 J a • I

+5 J 3oI

+6 J 3 • 1

+IO J 2« I

+28 J a»i

+98 J 2 • I
+IOOI J a«i
+2662 J 3»1

+12345 J 3*1
+34567 J 2*1

+10001 J 3 • 1

+i234567J3«i

+7654321 J 3«I

+IOOOOOI J a«i

+99009901 J 3»I

-9 J 3 • I

+9 vl 3 ♦ I

+1 4641 J 3» I

+ 513 J 2 •!

+96 J 2 • I

4. ^ <<h iK ^ 4< 4.

Pig. 4.5 Print'out of an input tape for 'Special Factorize' .

In view of the crudeness of the process used it might be thought that the programme would be very
slow. This is not so, however. For most of the numbers shown the factorization takes a hardly per-
ceptible time and the output punch is running nearly continuously. For 'difficult* numbers having large
prime factors the running time can be easily calculated since It takes about 7 milliseconds to try each
value of d. Thus all values of d less than 1000 (i.e. 3, 5, 7, .... 999) can be tried in about 3^2
seconds, and it therefore takes at most this time to factorize any number up to 1000000. The number
99009901 shown takes about 12^ seconds. t

4.5 Another complete programme

We shall now -give an example of a programme using several subroutines. Suppose we have to tabulate
the values of

_ 7'/ y^ -1 . 3+2^2

for X = 0, %, 1, iMs, 6^.tt Since some of the quantities concerned cannot be represented directly

as fractions in the range -1 to +1, we must manipulate the formulae to a certain extent and introduce
scale factors. Instead of storing * we shall use the fraction ^ = 1/8 x, which is always in range (a
possible alternative would be to store 2x as an integer). The quantity y attains its largest value, of
rather less than Z% when x = &h ^o we can safely evaluate 1/4 y without overflow; we shall write 77 = 1/4 y
The value of y is a minimum, y = 1, when a; = 1 so that

i- < T) < 2.

=5, 7)

\ This number is actually a divisor of 10 +1 and more refined processes can be used on such numbers.
In this case, for example, it is known that any factor must be of the form 20fe + I, so we are actuallj
trying ten times too many trial factors. With general numbers It is easy to omit multiples of 3 and
5, which roughly doubles the speed of factorizing.

tt This can usefully be written x = 0(i4)6i4, meaning that x takes values from to 6i4 in steps of Ms.

- 74 -

SOME SIMPLE PROGRAMMES 4.5

13/5/60 6 Date & Serial No. j Printed during

I input of the
SPECIAL FACTOR I ZE Name of programme J programme tape.

430 »ax 4X3x5x7 4^ — Test number

TEST NUMBERS -« — Name of input tape

4. » a X 3

5 " 5

6 = 3X3
10 « a X 5

38 » a X 3 X 7
98 = a X 7 X 7
1001 »7XiiXi3
S663 « a X 1 1 X 1 1 X n

"345 = 3 '^ 5 X 833

34567 « 13 X 3659

loooi « 73 X 137

1334567 « 137 X 9731

7654331 » 19 X 403859

loooooi « 101 X 9901
99009901 = 3541 X 37961

9-3X3

14641 = II X 11 X 11 X II

513 -3X3X3X3X3X3X3X3X3

96 -3X3X3X3X3X3

Pig. 4.6 Output produced by the 'Special Factorize' programme.

If we substitute x = 8^ and y = 47] in the formula for y we get

_1_ 4. A2

128

77 = — ■ . (2)

32 ^

and we can see that the numerator and denominator and the result are all within the permissible range.
In terms of T) the formula for 2 becomes

116772 - 1 1^2 _ X
' ^ ijl6772 + 1 = i^rj2 + ^ • <3)

Since 1/16 < 772 < 77 < 7/8 both numerator and denominator are within range and are never negative.
Clearly also the numerator is always less than the denominator. We shall therefore deal with the
formulae (2) and (3) rather than (1).

There are 14 values of x and we shall therefore arrange the printing of the results on 14 lines
and in two columns; on each line there will be a value of x and the corresponding value of z. We
shall write a special subroutine for printing x (= 8^) preceded by CRLP, given the value of ^, since
this is very simple. t The reader should have no difficulty in understanding this subroutine, which
is as follows. The value of ^ is held as a fraction in J4, and no rounding is needed since ^ is
represented exactly and printed in full.

* Some library subroutines could be used for this but it would be inappropriate at present to describe
them in the necessary detail.

- 75 -

4.5

SOME SIMPLE PROGRAMMES

0.0

B 4

(30) 6 40

16 6 10

(13) 6 40

16 6 10

(a) 7 40

4 7 20

16 6 10

(12) 6 40

16 6 10

@ 6 40

6 7 20

16 6 10

0.7 1 10

' print CR LP

(pq)m = ^ = ■«
print integral part of x
' print decimal point

> print first digit of fractional part

plant link in 0.7

dummy (then obey link in 0,7)

The brief specification of this subroutine is:-

Print 8C(4) to one decimal (unrounded) preceded by CR LP.

Cue:

4 [0}72
0.0 60

Uses: f/0; X6, 7.

Link: XI. obeyed in 0.7.

Note: C(4) must be non-negative.

So that we can print z we shall suppose that a subroutine is available with the following brief
specification.

Print C(7) as a fraction (unrounded) preceded by Sp.

Cue:

51 H 72
0.3 60

Uses: UO; BO.

Link: ^1, obeyed in 0.7.

Programme-parameter in ^2 specifies the number of digits to be printed after the decimal
point. The sign is printed as Sp or a minus sign.

Since we have to evaluate a square-root we shall assume that the following subroutine is available.

Put into X6 the square root of the double-length fraction in X6 and 7 (or, briefly, p' = V'(pq)).

Cue:

74 |0l 72
0.0 60

Uses: UO; X5, 6, 7.

Link: XI, obeyed in 0.7.

There Is a loop stop if (pq) is negative.

- 76 -

SOME SIMPLE PROGRAMMES

4.5

We shall construct a special subroutine to evaluate z f rom <f according to the formulae (2) and (3)
above. It is convenient to keep the current value of ^ in ^4 during this programme; this subroutine
therefore takes ^ in X4t and places 2 in ^7 (ready for printing). There are a few points to watch.
First, the subroutine will itself have to call in a subroutine to evaluate a square root; it will
therefore have to remove the link from XI and preserve it until it is needed. Second, although ^ is
stored exactly (in Jir4), 7) will have to be represented approximately, since it is calculated from the
formula (2) above. Consequently the minimum value of rj^ may not be exactly 1/16 (as it should be)
and the numerator of the fraction in formula (3) may go slightly negative when it should be zero.
Since we do not wish to introduce spurious imaginary numbers we must make sure that the numerator in
formula (3) is replaced by zero if it goes negative. Apart from these points the subroutine is simply
a straightforward evaluation of formulae (2) and (3). We suppose the subroutine to occupy B5 and 6.

0.0

B 5

[+0.03125 ]

+0. 0234375

+0.0625

[El72

1.7

1 10

4 20

0.1

6. 01

0.0

4 11

0.0

6 25

7 21

7 00

0.2

6 01

0.2

7 03

= 1/32, becomes 1/32 + ^
= 3/128

= 1/16

read next block of y

count down from n

new element z^ to Jl

write Zj (or stop if OVR set)

increase 3ff to next element of z

count down from m

The running time for this sequence is about 18 seconds.

If the matrix is stored by columns instead of by rows then the address of a^j is

Aq + i + nj .

There is now no advantage to be gained by bringing in A by blocks because a 72-order will transfer
elements of the same column and we need them by rows. If we use single-word transfer orders to read A
and also to write the product vector z, we can use a sequence as follows:

Set (Aq, ) in. say. register k
Set (Zq, m) in X2

-♦. Set initial conditions
Set (Yq, n) in X3

D72 3 Read Y

Copy content of register k into A'4
Increase modifier in Jl: by 1

{0 gj70 4

Operate (modified by 3m)
3 66

^72 3
►Increase im by m
- 3 67

10 0171 2
Increase 2j(^ by 1
2 67

Read A

Read Y

Write Z

Inner
Loop

101

5.7

MODIFICATION

Note that %, which is used to pick out the elements of A, jis increased by m each time the inner loop
is traversed; in this way we can select the successive elements in a row. When the inner loop has been
traversed n times we have evaluated one element of z and % has been increased by mn. Before we can
calculate the next element of z we must set 4j^ to the address of the first element of the next row of A;
we can do this either by subtracting mn~l from it or, as we have done here, by storing it (in a register
k) before entering the inner loop. In this example we use the same values of the constants as in the
previous one. The ordinary register 0.0 is the "register k"; its content is initially (Aq, 0).
Register 0.3 is used to hold a useful constant when the order-pair in it is no longer needed. On the
programme sheet these two words are written in brackets, the conventional way of indicating that they
get changed during the course of the programme.

Initially (Aq,0). This is register k;
0.0^ is increased by 1 in the outer loop.

1073

- -0 0.

0.0

699

- -0 0.

810

- -0 0.

0.2

5 00

[ @

5 52 1

0.3

5 10

0.2

2 00

1— ■ ir
5

0.1

3 00

jl]72 3

0.0

4 00

5 00

0.0

5 11

► (J^Of n)

. (Zq, m)

C(5) = (Zq, m)

Ih^

10 0170 4
5.0 1 22 3
1.2 3 66

11]72 3
0.3 4 01
1.0 3 67

6 1 00

1.0

|0 0|71 2
35 2 01
0.5 2 67

(fast shift order)

C(5) = (m, 0) J

O.dj^ - m 2

Set (Zq, n) in X2

(pq) = ViG (rounding constant)

Set (Yq, n) in J3

First block of y to f/5

Set 4^ to first element of a row of A

► increase 0. 0^ by 1

aij to ^1

(pq)' = ipq) + aijyj (operate)

read next block of y
increase 4^ by m
count down from n

► write z-

Increase 2^ by 1
count down ' from m

Inner
Loop

The running time for this sequence is about 33 seconds.

It is sometimes necessary to work backwards through a set of locations in the main store; this
is to be avoided, if possible, since it is very easy to make mistakes. If we have to work backwards
through one or more sets of numbers it is simplest, if computer time and storage space permit, simply
to reverse the sets concerned, i.e. to copy them backwards, into a new set of locations, before per-
forming any arithmetical operations on them. Alternatively we may use only single-word transfers.

Suppose, as an example, that a set of n numbers starting in location Aq is to be copied backwards
into a set of n different locations starting at Bq. This means that: -

- 102 -

MODIFICATION

5.7

On- I is to be copied from Aq + n-l into Bq
o„_2 " " " " " ^0 + n-2 " Bq + 1.

II II II

An + 1

J5o + n-2
Bq + n-1

If block-transfers are used it is best to transfer the elements in the order indicated above. Suppose
that the constant (Aq, n) is stored in B13.7 and that the constant (Bq, 0) is stored in B17.4. The
simple method of using single-word transfers to read in the elements of set A will be illustrated first.
We must construct a modifier equal to the address of the last element of A, this is Aq + n-l. The
programme will read; -

113 7170
1 4 00

Q) 4 43

(25) 4 52

1 4 01

0.0

1
2

1 17 4 1 70
1 5 00

34 5 03
1 [5]72 5

10 0170 4
5.0 1 10 5
0.7 5 66

[5]73 5

1 [5]72 5
35 4 03

0.4+ 4 67

1.0

1 [l73 5

(Aq,' n) to XI from B13.7
C(4) = (Aq, n)

C(4) = (n-l. 0)

C(4) = (Aq + n - X. n)

(Bq, 0) to XI from B17.4

Bo to 5nf

5^ = Bo - 8

First block of fc' s into C/5

One word of set A into XI

Copy this into U5

Advance and test 5j|^

Write block of b' a

Read block of 6' s

Reduce 4^^^ by 1

Count down from n in 4^

Write last block of b' s

Since there is a single-word transfer order in the loop of the above programme it will be
relatively slow (one number every 16 milliseconds). This may not matter. A rather faster programme
may be written using block- transfers only. The following sequence will reverse the elements into
their new places (it is assumed that the new locations are quite distinct from the old).

103 -

5.7

MODIFICATION

1 ■ ' ■ 1

!»•

0.0

113 7170

1 4 00

(25) 4 52

1 4 01

117 4 170

34 1 03

1 |4|72 4

1 [5j72 1

4;0 6 00 4

5.0 6 10 1

0.7 1 66

|5|73 1

1 |5|72 1

35 4 03
1.0+ 4 66

w
7

1.0

LiJ72 4
35 4 03

0.4+ 4 67

1 |5|73 1

U^, n) to XI from B13.7

n ~ 9 to ^ff

C(4) = (Aq + n - 9. n)
(Bq. 0) to XI from B17.4
C(l) = (Bo ~ 8, 0)
Last block of a' s to f/4
First block of 6' s to (15

* Copy one number

Set

modifiers
^ and
dounter

' Work forward through b' s

4^ left unchanged but jump
occurs if 4p /

New block of a's if 4p =

Reduce 4^ by 1

Count down from n in 4^;

Write last block of 6' s

Note that the modifier 4 has the initial value A^ + n - 9; this is in fact 8 (or 1.0) less than the
address of the first number to be handled, which is necessary when working backwards through a set with
block- transfers just as it is when working forwards if we are writing and not simply reading. The 66-
order in 0.5+ causes no Jump when the position-number 4p is zero, i.e. just before the block-number is
reduced.

The administrative orders in a loop which works backwards through the n numbers of a set A (using
block- transfers) can be written down in abbreviated notation as follows:

Set (A(. + n ~ 9. ) in X2

1 072 2

Read A

-►Operate (modified by 2^^)

35 2 03
2 66
CI]72 2
■>35 2 03

Read A

(67)

104

MODIFICATION

5.7

As before, the dots indicate the orders concerned with manipulating any other sets of numbers. Here the
initial setting of 2j,^ Is 8 (or 1.0) less than the address of the first number to be handled, i.e. the
last number of the set. The first 72-order reads in the block containing this number. The 66-order and
the 03-order preceding it can be regarded as leaving 2]^ unchanged but sensing the moment when the last
number of the set in the computing store (i.e. that in position of the block) has been dealt with. The
modifier is reduced by the 03-order after the 72-order.

If we are writing (or writing and reading) the numbers of the set the orders are as follows.

Set (An + « - 9. ) in X2

1 072 2

Read A

-> Ope rate (modified by 2^)

35 2 03
2 66
1 n73 2
072 2

Write A
Read A

>35 2 03

(67)

1 073 2

Write A

Note that in this sequence the modifier has the same starting value as in the previous sequence. The
final 73-order outside the loop writes back the block containing the first number of the set.

As an example we will suppose that the n coefficients a^, og, og, ...» a„ have been computed and
stored in consecutive locations with

a^ in location A^,

a^ in location A^ + 1,

a in location A, + n - 1.

We have to evaluate the n + 1 quantities Xq, x^, x^, ... *„ from the following recurrence relations:

*n = y-

^n-2 = °n-l ^n-1 +^*n'

'n-3

^n-2 *n-2 *n-l'

Xq - a^Xi + *2-

The values of the x' s are to be stored in consecutive locations with xq in location Xq in the Main Store,
arj in A'o + 1 «„ in Xq + n. The following constants are specified in B9: -

B9.0 contains (Xq, n) ,
B9.1 contains (A^, 0),
B9. 2 contains y.

If overflow occurs a special overflow routine kept in U3 is to be called in by jumping to 3.0.

We need two modifiers and, since we shall have to work backwards through both sets (the a' s and the
x'a), both modifiers must be 8 less than the addresses of the first numbers to be handled (this is so
whether or not writing transfers are concerned). One modifier must be initially 8 less than the
address of a„, i.e. it must start at the value Ai + n - 9. The other modifier must be 8 less than the

105

5.7

MODIFICATION

address of x^, i.e. Xq + n - 8. In the following sequence of orders x^ _ ^+j is in Jf3 and x„ _ ^+2 is
in XQ when the orders are encountered for evaluating x^_^.

0.0

1.0

9 [4] 72
4.0 4 00
(j) 4 43
@) 4 52

4 5 00
4.0 4 01
Q) 4 41
4. 1 5 01
35 5 03
4.2 3 00

1 [4] 72 4

1 [5] 72 5
4.0 3 10 4

6 00

1.4+ 60

33 7 00

5.0 3 22 5

3.0 65

4.0 6 10 4

3 6 00

4,0 3 00 4
35 5 03

1.4 5 66
[5172 5

35 5 03

35 4 03

1.6+ 4 66

1 [4)73 4

[4]72 4

35 4 03

0.7+ 4 67
1 Rl 73 4

Constants to t/4

C(4) = {Xq. n)

C(4) = a^. n - 8)

4^ = n - 8

5;V = n - 8

C(4) = do + n - 8, n)

C(4) = ^0 + n - 8. n + 1)

C(5) = (4q + n - 8, 0)

C(5) = (^0 + n - 9. 0)

y = x„ to J3

Block containing location for ;c„ to {/4

Block containing a^ to (75

;t„ to m

"x . ," = in ^6

Enter cycle

^ to XI

^n-r ^ '^n-r + l ^n-r + 1 "^ ^n-r + 2 ^° ^^

Jump if overflow
x^_^ to C/4

■^n-r + l ^^-^ ^6

X to X3

n—r

ready for calculation of A:„_r-i

> Read in new a' s as needed

>• Store %' s as calculated

Count down from n+1 in 4,

Store last block of x' s

If speed is an important consideration in a loop consideration should be given to the possibility
of completely or partially "unrolling" the loop. In this technique the orders required are written out
one after the other instead of in the form of a loop; they are then often unmodified. Speed is gained
because many of the orders concerned with counting or changing modifiers are not encountered every time
the operate orders are obeyed. This device is chiefly useful when the operate orders are few in
A number, and when the sets concerned fit neatly into blocks.

106

MODIFICATION

5.8

5.8 Modification of other orders

We have so far described how the following kinds of orders are modified.

(a) The arithmetical orders (Functions 00 to 27), in which the position part of the modifier is added
to the Af- address in the order.

(b) The block- transfer orders (Functions 72 and 73), in which the block part of the modifier is added
to the first address written in the order.

(c) The single-word transfer orders (Functions 70 and 71), in which the whole of the modifier is added
to the address (represented by the N- and X- digits) in the order.

These orders are modified chiefly in loops or cycles of orders in which the modifier is increased or
decreased (usually by 1) every time the loop is traversed. The modifier is usually best regarded as a
main store address.

The other orders in the Pegasus order-code can also be modified, but usually for different reasons
and hardly ever in a loop (except incidentally). With these orders it is usually best to regard the
modifier simply as an integer.

The orders of groups 4, 5 and 6 are modified by having the 10 least-significant digits of the
modifier added to the first address before the order is obeyed. Usually the other digits of the
modifier (i.e. the 3 most-significant digits) are zero. For example, suppose C(4) -- (12. 243), so that
4|f = 12; "then the order

3) 7 50 4
will have the same effect as the order

^ 7 50
and C(7) will be multiplied by 2i5, With the same modifier the order

@ 5 40 4
will have the same effect as the order

22) 5 40

and the integer 22 will be placed in X5.

As with the transfer orders (Sections 5.3 and 5.4), the ^'-address is extended inside the order-
register by 3 binary digits at the instant the order is obeyed. The effective value of A^ may thus go
up to 1023. Any carries beyond these 10 binary digits are disregarded.

As an example, suppose the number y in ordinary register 5.0 is to be shifted down arithmetically
by h places. The quantity h is equal to an integer stored in 5.1, supposed non-negative. The
following orders will do this.

5.0 6 00

y to X6

0.0

5.1 7 00

h to XI

@ 7 52

h to 7^

(o) 6 51 7

shift y down h places

5.0 6 10

result to 5.0

An important application of modified group 4 orders is to copy a number from the modifier of one
accumulator into the counter of another, assuming that the modifier does not occupy more than 10 binary
digits. On the whole it is rare to use modified orders of groups 4 and 5, and many of the applications
can be considered as tricks.

Modified jump orders, although only occasionally needed, are important; when using them it is
usually necessary to know the way in which the A^-address of an order is represented in the computdr
(see Sec. 3.12). An important special case of the use of a modified jump is provided by what are known
as computing store links.

Suppose we have a small group of orders or a subroutine which starts in UZ.O. We wish to jump to
3.0 to enter the subroutine and we must arrange that the subroutine returns control to the appropriate
place in the computing store when it has finished. This is normally done by setting a link in XI but
this is inconvenient if we wish to call in the subroutine frequently during the course of a short
calculation. As we mentioned earlier, most subroutines use as working space some of the ordinary
registers originally occupied by their own orders and constants; they do this, of course, only after
the original contents of the registers are no longer required. This is permissible in a computer with
a two-level store, such as Pegasus, since a fresh copy of the subroutine will normally be brought into the
computing store every time it is needed. In some programmes this may waste some time because of the
need to wait for the drum on most block- transfer orders. In order to avoid this it is possible to use
self-preserving routines which may be entered repeatedly after they have been transferred only once.
Such a subroutine must be so written that its orders are not spoilt by its own operation; for example
by using only accumulators as working space, or by resetting itself (or perhaps only its first block)
in the computing store before exit. It is often useful to design such a subroutine so that its last
order is a modified jump order. This is most easily described by an example.

-107 -

5.8

►DIPICATION

Suppose the subroutine has already been transferred to US and £/4 of the computing store and it is
to be entered at 3.0. Suppose further it is desired to call it in from 0.0 in a master-programme and
that return to the master-programme is to be to 0.3+. The cue is written in the master-programme as
follows.

0.0

(0.3-H 1 40
3.0 60

set link for return to 0.3+
enter subroutine

The first order of this cue is written in a rather unconventional way; when it is obeyed its A^-digits
are placed in XI. The actual digits, however, need not be known (they are specified in Sec. 3.12). The
second order of the cue is simply a jump to the subroutine. Note that these two orders need not form an
order-pair.

At the end of the subroutine there may be the following two orders: -

25; 1 52
60 1

address to %

return to raaster-prograjnme

The first order here simply shifts up into the modifier position in XI the digits originally put in by
the first order of the cue, i.e. the digits of "0.3+". The second order gets these digits aiided to
its iV-address since it is modified by 1^; the jump consequently has an effective address of 0.3+,

The main advantage of this technique is its ease of use, we need only two orders to set the link
and call in the subroutine. Computing store links are not restricted to self -preserving subroutines;
for example we may be able to enter a subroutine by a group of orders similar to the following.

0.0

/""N

KlS ^ ^°

28 jo] 72

0.1 60

set link for return to 2.4

*■ normal cue

Note that the subroutine must be specially written to accept such links; this would be explicitly stated
in the specification of the subroutine.

Sometimes a subroutine may be so written that it can use either an ordinary order-pair link or a
computing store link of the kind just described. These can easily be distinguished if the order-pair
link is a go order-pair, as it nearly always would be, since it will then appear negative because of the
1 in the stop/ go digit (see Sec. 3.9). A computing store link will appear non-negative since it was
set by a 40-order. The terminating orders in the subroutine may therefore be as follows.

jump if computing store link
plant and obey order-pair link
address to %

exit

may be overwritten by
order-pair link

Note that this sequence is not self-preserving, and that it spoils C(l) if it is a computing store
link. There are ways of avoiding these disadvantages should it be thought advisable.

The multi'way switch is another useful programming device which is made possible by modified
jump orders. We have already seen how the sequence of orders obeyed by the computer can be inter-
rupted by a conditional Jump order so that one of two alternative sequences can be entered. If there
are more than two alternatives we can often select the appropriate one by using two or more conditional
jumps. For example, suppose that an integer a has just been calculated and placed in J6; it may
have the values -1, 0, 1 or 2 and it has been arranged that the subsequent course of the calculation
depends on the value of a (in fact a may have been artifically introduced specially to discriminate
between the various possibilities). A possible method of selecting the appropriate sequence is to
use the following orders.

— ^
0.0

jump to 2.0 if a = -1
jump to 2. 5 if a =
jump to 2. 2+ if a = 2
jump to 3.0 if a = 1

108

MODIFICATION

5.8

In a sequence of this kind the addresses in the jumps can be quite arbitrary. If, as often happens,
the several sequences are too long be held simultaneously in the computing store, then the jumps can
lead to cues for calling them In.

By using a modified jump we can get a similar effect, for example:

a to 6j(^

jump determined by a

Here the jump will go to 4,0 if a = -1, to 4. 1 if a = 0, and so on; and we can write the cues we need
in these registers. An array of cues like this is often referred to as a cue-directory. Another
possibility is to store the various sequences to be called in in consecutive blocks in the main store
and to enter them with a modified cue or block- transfer order, thus for example:

ffa

(25) 6 52
4. 1 60 6

0.0

___

0.3

@ 6 52

0.4

21 [0] 72 6
0.0 60

a to 6jj

block 21 + a to f/0

enter new block.

Here the quantity a is shifted up into the block part of the modifier in XQ and used to modify a 72-
order, which reads B20 into f/0 if a = -1, or B21 if a = 0, B22 if a = I, and so on.

We have now to describe how the remaining orders of group 7 are modified. It is seldom that any
of these need to be modified in a basic Pegasus installation. The following summarizes the effects.

(a) the orders 70, 71, 74, 75 are modified by having the whole of the modifier added to the iV- and X-
digits written in the order,

(b) the orders 72, 73, 76, 77 are modified by having the block part of the modifier added to the
JV~digits written in the order.

Apart from the main store transfers, only the 74-order can usefully be modified in a basic installation
and the 75~order is unassigned. The 76-order is often modified when magnetic tape or cards are fitted;
this is described in Chapter 10. The addresses in a 77-order do not affect the operation of the order.

The way in which the various orders are modified can be conveniently summarized in the way shown
in Pig. 5.2. Here the top line represents the order being modified; the lines below represent the
modifier, of which the shaded part is added to those digits of the order which lie above them in the

ORDER BEING
MODIFIED

N
U P

00-27

72,73

70.71,74

40-67

FUNCTION

MODIFIER:

SHADED PART
IS ADDED
TO ORDER

(Modulo 4)

(Modulo 1024)

Pig. 5.2 How various orders are modified.

109 -

5-8 MODIFICATION

figure. Note that the order is extended on the left by 3 binary digits at the moment it is modified and
just before it is obeyed. Any carries which occur beyond this will be disregarded, i.e. only the 10 least-
significant bits of the modified /V- address are relevant. In the jump orders and the arithmetical
orders, i.e. in the orders of groups 0, 1, 2 and 6, it is only the 7 least-significant digits which
are relevant.

The function part of the order is never affected by modification, and the JT-part is modified only
in the single-word transfer orders (70 and 71) and the external-conditioning order (74). If unconven-
tional methods of modifying are ever needed (for example to change the function of an order) this can
always be done by adopting a system often used in computers not equipped for modification; this is to
change the order in Its ordinary register by adding something to it, for example by using a 11- or 13-
order. This technique is very seldom required and would normally rank as a trick.

We sometimes have to refer to a list or table of numbers or other data held in the store and
extract a particular entry from it. This is usually referred to as a table look-up operation. Normally
the table will be held in the main store and often each entry will occupy one word; if this is so we can
extract any particular entry, say the n-th. by putting n into the modifier position of an accumulator and
then obeying a modified single-word read order. For example, if n is held as an integer in 5.0 and the
table is so stored that the entry corresponding to n = is in S94.0. then the following orders will put
into XI the entry corresponding to the specified value of n.

5.0

1 00

n to 1q

1 52

n to 1^

|94

j]70 1

n-th en

If each entry in the table occupies more than one word it may be advantageous to use a block-transfer.
For example, suppose each entry occupies two words and the table starts in B94. 0, then the following
orders will place in X<i and 5 the two words corresponding to C(5.0).

5.0 5 00 n to 5^

26; 5 52 2n to 5^^

94 [4]72 5 block containing entry to (/4

4.0 4 00 5 first word to J4

4. 1 5 00 5 second word to ^^5.

Looking up such a table Is particularly simple if each entry occupies a whole block.

Often the entries in the table will be small integers or other items which can be specified
adequately by using only a few binary digits. We can then pack several entries into a word and make
the table more compact. As an example, suppose there are three entries in each word, each occupying
13 bits. Suppose we denote the n-th entry by /(n) since it corresponds to the mathematical idea of
a function of n. It is customary and convenient to number the entries in the table from zero, so that
the first entry is /(O) and not /(I); to avoid circumlocution we shall talk of the n-th entry when we
mean /(n). To find out the number of the word in which /(n) is stored we divide n by 3; suppose the
quotient and remainder are q and r, respectively, so that

n = 3g + r.

The word we want is the qr-th (starting from 0) and r (which will be 0, 1 or 2) indicates which part of
the word contains /(n). For example, the 7th word in the table (or rather word no. 7, starting from
no. 0) will contain the values of /(21), /(22) and /(23), packed in the following way.

7th word
of table

13 bits

/(21) /(22) /(23)

If n is the non-negative integer in 5.0 then the following sequence will place /(n) in the left-hand
third of XI.

110

MODIFICATION

5.9

0.0

- -0 0.

127

7 60

5.0

7 00

®.

6 40

7 52

|94

_0]70 7

0.0

6 20

1 52 7

0.1

1 05

► = 13.2"

• collating mask l^-' 0^6

n to XI

divide n by 3,
' q to XT, r to X6

q to 7^

g-th word to XI

13r to If^ (= 0. 13 or 26)

shift word up 13r places

replace all but left-hand third by zeros.

If the result is wanted in the right-hand third of the word instead of the left-hand third, we can omit
the collating mask and replace the last order of the sequence by a logical shift down of 26 places;
this is, however, relatively slow. A faster alternative is obtained by putting -13.2'^^ in 0.0 so that
the order in 0.4+ puts -13r into 7^ (actually modulo 8192); the modified shift order should then be
written as follows.

26; 1 53 7

This order will then shift the word down 26-13r places. If a still faster process is wanted we can
replace the division by 3 by a multiplication by 1/3, or by 2' ^-^73 (rounded up in either case).

A table look-up process is most useful when the entries in the table are not related in any simple
or systematic way to the values of n. In fact the entries can be quite arbitrary. Sometimes it is
possible to calculate from the value of n just what the corresponding entry is. If the time and
complexity of this calculation are not too great then it may well be better to perform it instead of
looking up a table. The programmer should always remember that looking up a table is often time-
consuming (especially if Interpolation is required) and that the table may occupy a large amount of
storage.

Code conversion on input or output is an important application of the table look-up technique; a
useful way of doing it is described in Sec. 6.3.

There is another kind of table look-up in which the entry to be extracted is determined by the
entry itself and not (as In the processes described above) by where it is stored. This usually involves
searching, and may therefore be referred to as a table searching process. In a typical application the
entries may be word-pairs, and we require the word-pair whose first word (the keyword) is just greater
than a specified number, the word-pairs being stored in order of magnitude of the keyword. Alternatively
the entries may be blocks. In such a table not all possible keywords are present. One way of finding
the appropriate entry is to determine first in which half of the table it lies, this can be done by
examining the keyword of the central entry. We next look at the keyword of the central entry of this
half of the table to find out the appropriate quarter of the table, and so on. In other words we
proceed by repeated bisection of the table. t Special care has to be taken if the number of entries is
not an exact power of 2. There are a number of ways of making the process faster. For example, we can
store separately, in special blocks, copies of some of the keywords, grouped together so that the first
few keywords we have to examine can all be brought into the computing store by a single block- transfer.
In certain tables the distribution of the keywords may be such that it is possible to calculate approxi-
mately where a given keyword is to be found and so to narrow the search.

5.9 A complete programme using modification

We shall now describe a complete programme illustrating the application of modified orders and the
use of subroutines. It is not intended to be suitable for practical use.

The programme is designed to read in a paper tape on which are punched account numbers, quantities
and prices. Fig. 5.3 is an example of the print-out of an acceptable input data tape (see Sections 4.3
and 4.4). Here the first column contains the account numbers, the second contains the quantities, and
the third the prices in sterling. The account number may be from to 99; the quantities and prices
may not be negative. Any number of lines may appear; after the last line the "account number" minus 1
is punched (as -1) to indicate the end of the tape. As the tape is read in by the programme a value is
found for each line by multiplying the quantity by the price; this value is added into a total
corresponding to the account number. When the end of the tape is reached there will therefore be up to
100 total values, one for each account number that has appeared on the tape. These totals are then
printed out and the computer stops.

t This is sometimes called a Weierstrass bisection process, after the famous 19th Century mathematician
who first applied it to mathematical analysis. It Is also known as a logarithmic search since the
time needed goes up as the logarithm of the number of entries in the table.

5.9 MODIFICATION

12 6 x*t $9to
49 144 36»I3» o
90 30 zo*io« o
21 S3 6» 7, 6
IS xa a«£ 4» o

5 10 3« 6,ii

90 xo 51 « O9 o

13 3 4« o» o

3£ 2000 £• !» 3

12 24 i6»io« 6

-J

Pig. 5.3 Print-out of a data tape.

The process and the programme can best be described with the aid of a flow diagram, see Pig. 5.4t.
At the start 13 blocks of the main store are cleared; these contain 104 storage locations, of which the
first 100 are used to receive the totals as they are built up; the other 4 locations are not used.
Next the input loop is entered: this starts at link-point 1. A subroutine (P) is used to read integers
from the data tape; it Is called in twice during the input loop, the first time to read in the account
number and the second time to read in the quantity. These numbers are tested as soon as they have been
read; if the account number is negative then the end of the tape has been reached and the computer ,
leaves the input loop and goes to link-point 9 as described below. If the account number Is between
and 99 and the quantity is not negative the input loop continues by calling in another subroutine (Q) to
read the price in from the tape. This price Is multiplied by the quantity to form the value, which is
then added into the total corresponding to the account number. The computer stops, however, if either
the value or the new total exceeds the capacity of a register (about £1100 million since sums of money
are stored as integral multiples of a penny). Control is then returned to link-point 1 to repeat the
input loop.

Link-point 9 is reached when all the data have been read in; here two line feeds are printed (to
separate subsequent printing from other printing on the page), after which a counter (c) is set equal
to 100 and a modifier (n) is set equal to zero; the output loop is then entered at link-point 2. Here
the totals are examined successively to see if they are zero. If any non-zero total is found a jump
occurs to link-point 3, where subroutine A is called in to print a carriage return, a line feed and the
account number (n); subroutine D is then ased to print "a space followed by the value of the total.
When all 100 totals have been examined the computer stops (at no. 8),

The programme tape contains only the master-programmes. The four subroutines used (A, D, P and G)
are assumed to be already in the main store; they are included in the Lesser Library and their specifi-
cations are given in Appendix 2. This small collection of subroutines can be put into the main store
(in fixed locations) at any time by running in a certain tape. In order to read in the programme tape
it is placed in the main tape-reader, and the Initial Orders are called in by operating the Start key
(see Sec. 4.3). The Initial Orders then read in the tape; which starts, as usual, with a D-directive
to cause printing of the date and serial number as the tape goes in. After this is an N and the name
of the programme - ACCUMULATE TOTALS. Then come the four blocks of the master-programme, which are
stored by the Initial Orders in B2 to B5 in the main store. At the end of the tape is punched the
directive E 2,0, and a few "Erases" (used to mark the end of the tape). As soon as the Initial Orders
read E 2. there is a 77-stop; when this occurs the programme tape is taken out of the tape-reader and
the start of the data tape is placed in the tape-reader. The Run key is then operated and the first
four blocks of programme, B2 to ^5, are copied by the Initial Orders into the computing store (this is
actually the whole of the master-programme), and a jump then occurs to the a-order in 0.0. In this
way the computer starts to obey the master-programme.

The master-programme is given in full in Pig. 5.5; in order to facilitate study the numbers of the
link-points and stops in the flow-diagram have been inserted on the programme sheet against the
appropriate orders. At the moment when the computer starts to obey it the whole of the master-programme
is in the computing store and the data tape is in the main tape-reader. The first four orders are those
which clear 13 consecutive blocks in the main store, starting with BllO, to receive the totals. This is
done by copying U6, which contains eight zeros, into each block in turn (see Sec. 3.10). Note that 2q
is initially zero and is increased by 1 each time the loop is traversed.

When the 13 blocks have been cleared the input loop is entered (link-point 1, 6-order in 0.2); this
loop consists of the orders from 0.2+ to 2.1 inclusive. The first few orders are those which call in
subroutine P to read in the first number on the data tape; this subroutine uses UQ and Ul and leaves
the account number (n) in X6 before obeying the link (1), which restores B2 in f/O and jumps to 0.5.

t This flow-diagram Is written according to one of the many sets of conventional rules which have been
devised. The operations are all written in operation boxes, which have one entry arrow and one exit
arrow. The tests are all written In ovals, which have one entry and two exits. Stops and Starts are
indicated by large circles. The link'points, which are here numbered for convenience with the stops
and starts, are the small circles; all the link-points bearing the same number can be thought of as
joined together. A link-point can have any number of entries but may have one exit only; the
numbering Is useful for reference purposes. The arrows may lead only from one box (or link-point etc.)
to another - they may not Join up.

112

MODIFICATION

5.9

STARTS »

/^stopV

STOP

STOP \YES

Clear 13 blocks to receive
totals; starting at BllO

Call In subroutine F to
read account number (n)

(S>

Yes

YES

€

> 99?

)

Store n. then call in

subroutine F to read

quantity (g)

YES

(7^

Call in subroutine G to
read price (p)

Find value ~ g x p

Is value too large?

Add value to n-th total

~Ki)

(3>

-*0

Print two line-feeds

Set n = and c = 100

-A)

Read n-th total, = t

= 0?)

YES

Increase n by 1
Reduce c by 1

-dZZ)

YES

/STOP^

Call in subroutine A (cue 1)
to print CR LF&acct. no. (n)

Call in subroutine D (cue 2)
to print Sp& total value (t)

Pig. 5. 4 Flow-diagram of the programme "Accumulate Totals"

113

5.9

MODIFICATION

D
N
ACCUMULATE TOTALS

C(2) = (0,13)

Add 1 to 2b
Count in 2c

INPUT LOOP
set link (1)

Clear
13 blocks.
* starting
at SllO.

link (1)

Call in
subroutine

P
to read
account
number (n)
into X6

Leave input loop if n < 0.

n to X4:

n - 100

Restore f/1

B Stop if n > 99

n to ifi

1.1

1.0

1.2

(Ti*^

\zJ

4.0

1.5

(Pi

0.5+

1.6

4,0

r^ k

I 7Jil-_ , ■ ■•

1.6-t

61
70

|110

set link (2)
Jump to cue

■link (2)

Call in
subroutine
P
> to read
quantity (q)
into .Y6

« Stop if q <'0
Store q in 4.0

set link (3)
in Xe

■link (3)

Call in
subroutine
G
^ to read
price (p)
into XI

value = <j X p to JIT?
■ Stop if value too large
read n-th total to XI
add new value

1 -*-

©-

2.0

[JTo"

2
0.2+

16
16
100
110
4.0
3.0
2.6
110
2.4

2.6+

3.0

3.3

0.0

3.4

5
3.7

0,2

2.5

write n-th total
restore f/O
repeat

Print two
'line-feeds

C(3)

(0, 100)

First 8 totals to f74

^n-th total (0
to X5

Jump if f ;* 0,

OUTPUT LOOP

increase 3j^

}more totals
to f/4
1 count down
J from 100

Stop at the end of
the programme

C(7) = n(account no. )

programme-
' parameter = 2

set link (4)

►cue (1)

Uink (4)

1 prograrame-

J parameter = 10

t to XI

set link (5)

■cue (2)

.link (5)

Call in
subroutine

A
to print
CR LP and
account no.
(n) -2 digits

Call in
subroutine

D
to print
space and
total (f)
"10 digits
in £ column.
Then go
to0.

E 2.0

Pig. 5. 5 The master- programme of "Accumulate Totals

- 114

MODIFICATION

5.9

If the account number n is negative we have reached the end of the tape and there is a jump to 2. 1+;
but otherwise n is tested, to make sure it does not exceed 99, and is placed in 4^^ by the fast logical
shift for later use.

The first few orders in Ul call in subroutine P again to read in the next number on the tape (i.e.
the quantity q). Note that the cue is not written out again; we have simply jumped to 0,3 to save a
few orders. This time the link in XI is the order-pair in 1.1. which restores Ul and jumps to 1.2 when
the quantity has been placed in X6; if q is not negative it is temporarily stored in 4.0. The next
few orders call in another subroutine (G) to read the price from the data tape. This subroutine is a
little unusual in taking its link from X6 and leaving its result in J^l. t When it has been called in it
uses UO and Ul and reads the price, converting it to an integral number of pence in XI. The link
restores Ul and jumps to 1.6 where the value is found by a multiplication. If this value is a single-
length number X6 will be clear and the n-th total gets the value added to it. This is done by a kind of
table look-up, a modified 70-order extracts the n-th total from the location n ahead of BllO.O and puts
it into XI, the value is added to this and the result put back by a modified 71-order. If the new total
overflows the computer will stop at this point (see Sec. 3.10). After this UO is restored (Ul was
restored by the last link) and we jump to 0.2+ to repeat the input loop.

Eventually the "account number" minus 1 is read by subroutine F and the a-order in 0.5 then causes
a jump to 2.1+ (i.e. link-point 9) where two line feeds are printed. The programme now has to print all
the non-zero totals. First we put (0. 100) into X3; at a later stage 3j^ is the account number n and 3^
is the counter c which is used to stop the process when all the 100 totals have been dealt with. The
next order (the fc-order in 2.3) is

110 [4]72 3

In fact when this order is obeyed 3^ is zero so the order need not have been written as a modified one
at all. In other programmes the modification in this situation is necessary and it is therefore
customary to write a modified order, even though the modifier may be zero, as it is in this programme.
The effect of the order is to copy into Ui the eight totals corresponding to account numbers to 7.
The output loop is then entered at link-point 2, i.e. the a-order in 2.4.

The remaining orders in U2 are a special case of the simple standard loop given at the beginning
of Section 5.6. The operate orders are

4.0 5 00 3

3.0 5 61

these simply pick out the n-th total and jump if any printing is required. This loop will not
therefore be described further here. If the total is not zero we go to 3.0 (link-point 3); the
orders in t/3 are concerned with printing the account number n and the corresponding total t, using
subroutines A and D in the Lesser Library. To use subroutine A we must put n into XT, place in X2 the
number of digits (in this case 2), set a link and obey the first of the two cues given in the
specification (to precede the printing by carriage return and line feed). The first order in [^3 is

7 40 3

1/2/61 7

LESSER LIBRARY

1/2/61 -8

ACCUMULATE TOTALS

371308,

► 15.

46513884,

»I9»

8408594

» 16 •

34II94'

* 9*

1081 500033*

. 8.

775703*

>I0«

I7t<

► 13*

538451476.

.15.

2556865,

► 3«

75939*

► 19.

843870,

» i»

594461 666,

I3«

8o7io«

9»

16969791 ,

I3»

7070554,

763781 •

Pig. 5. 6 Accumulate Totals - Computer Output.

This has been done in case it is required to put the result into the main store with a 71-order.

- 115

5.10

MODIFICATION

and has the effect of copying the account number from 3j|f Into the right-hand end of XT. When subroutine
A has printed the value of n we call in subroutine D to print the total t (which is in X5) since we want
it printed as a sterling sum of money. Here we may have as many as 10 digits in the £ column so we put
10 in X2 as required by subroutine D (see the specification in Appendix 2); we use the second cue since
we wish the total t to be printed on the same line as the account number. When this subroutine has
finished it obeys the link written in 3.7, which causes a jump back to 2.5 in the main output loop. When
all the non-zero totals have been printed the 77-stop in 0.6+ is obeyed.

Figure 5.6 shows the computer output when the programme was run. The first two lines were printed
by the Initial Orders when the Lesser Library tape was put in; the first line being the date and serial
number printed because this tape has a D-directive on it. The third line is the result of the D-directive
at the head of the "Accumulate Totals" tape (note that the serial number is one more than that printed on
the first line). Next follows the name of the programme. The rest of the printing was produced by the
programme itself; one line of printing for each non-zero total (the data tape was not that shown in
Pig. 5.3). The first column contains the account numbers, and opposite each of these is printed the
corresponding total.

5. 10 Programming tricks

We shall now describe a few tricks, which may be defined as programming devices arising accidentally
from the design of the computer and for which no provision was deliberately made. Every computer has its
repertoire of tricks, though Pegasus has perhaps fewer than mostt, and programmers generally find it
satisfying to discover a trick or apply one to some process. On the whole tricks are to be avoided when
writing a programme since one can easily spend much time in devising them. Their use is legitimate if
the space occupied by a programme is important (for example, a certain section may have to be fitted into
a block) or if a great deal of time can be saved. If a trick is used in a programme then it should be
fully explained on the programme sheet so that someone unfamiliar with it (Including, probably, the author
after the lapse of a few months) can follow what is intended; such an explanation is valuable if, as
often happens, the programme has to be altered during development or at a later date. Some tricks may
make it exceedingly difficult to alter the programme and should not in general be used.

A useful device, which hardly ranks as a trick, is that known as a. binary switch. This can be
described as any technique for ensuring that certain effects occur on alternate cycles of a loop, or on
alternate times that a particular sequence is obeyed. The simplest method is probably to use the sign-
digit of an accumulator, this digit being reversed every time the loop is traversed. For example, suppose
C(2) ^ on entrr. -

" — 1^

I ^

* ^

. orders obeyed 1st, 3rd, 5th, ... times

2 62

32 2 06
2 67

. orders obeyed 2nd, 4th, 6th, . . . times

reverse sign bit in X2
count

The rest of the accumulator can hold, for example, a modifier and a counter. There are many other ways
of doing this; if an accumulator is available, for example, one can do it by counting or by using a
modifier which is given two values alternately. Occasionally we may require that a certain number
takes two different values on alternate cycles; here we can often use an 04-order or an 06-order in the
loop to change the number from one value to another (the loop written above is only a special case of
this). For example, suppose C(2) is to be alternately 19 and 43; before entering the loop we set the
value appropriate to the first cycle, and in the loop we have the order

6?) 2 44

which changes 19 into 43 or vice versa.

t This is not accidental. The computer was deliberately designed in such a way as to minimise the need
for tricks. Most of the orders carry out the kind of operations one wants, and most of the elementary
processes can be done with one or two orders used in a quite straightforward way.

- 116 -

MODIFICATION 5.10

Before passing on to the real tricks we shall draw attention to certain useful effects obtainable
in the ordinary way. Consider, for example, the order

116 51

This order will clear X6 only if its original content was or -1. The order

IT) 6 53

will clear X6 only if C(6) = or +1 originally. We can sometimes use these ideas to test for two
numbers simultaneously. For example, if C(6) is a small integer, the two orders

(T)^ 51

3.0 6 67

will cause a jump unless C(6) was +1 or +2.

The overflow- indicator can sometimes be used to store one bit of information. It can be set by
the order

32 02

or cleared (and tested) by a 64- or 65- order. The order

clears OVR and the sign bit in XI. The order

N 02

where N is any register, will set OVR only if the content of the register is -1.0; if iV = 15 we can
in this way sense whether or not the setting of the handswitches is with OT = 1 and HI to H19 clear.
We can sometimes avoid the necessity for counting in processing a list of numbers by arranging
that the last number is distinguishable in some way. For example, if all the numbers are positive we
can add an extra negative or zero number at the end; or if they are all small we can put in an extra
one which is very large. A useful number for this is the fraction -1.0 which can easily be detected
by a pair of orders such as the following, for example:

5.0 02 3

2.0 65

Suppose we have two modifiers and counters in 5.0 and 5.1 then, provided the sign-bits are the
same, the following sequence of orders will cause a jump only if the modifiers are equal.

5.0 6 00

5. 1 6 06

14) 6 50

3.0 64

We assume here that the counters are less than 2^^, and that OVR is clear on entry to the sequence.

It is sometimes necessary to mark or record the fact that a certain sequence of orders has been
obeyed or a certain fact has occurred. This may happen if, for example, two branches of the
programme join up and later the programme has to separate into two again. There are many ways of
marking; a link may be set which is later obeyed (as in subroutines), or the sign-bit or the
modifier or the counter in some accumulator may be given a certain value and later tested; perhaps
OVR can be set by one branch and not by the other. Note that three markers can fairly easily be held
in a single accumulator; the sign-bit can be tested by a 62- or 63-order and is the most convenient
one to use. The order

2.0 6 66
will cause a jump unless 6p = 7. The order

2.0 6 67
will cause a jump unless 6c = 1. The orders

@ 6 43

2.0 6 67

will cause a jump unless 6^; = 14, and so on. If the modifier in X5 does not exceed 1023 (or 127.7)
then the following group of orders will cause a loop stop unless the counter in 4. is equal to the
modifier in X5,

117 -

5.10

MODIFICATION

f ►

0.0

4.0 6 00
6 44 5
0.0 6 67

6c= 1 ^ 5^ - 4.0^
jump unless 5j|^ = 4. 0^

This group of orders might perhaps be classified as a trick; note that 4. Oj^ and 5^ do not affect the
test, and neither do the sign digits. The 67-order can, of course, jump elsewhere if a stop is not
wanted. One can sometimes usefully mark that a certain group of orders has been obeyed by replacing
an unconditional jump by a 66- or a 67-order which, under the circumstances, will always cause a Jump.
For example, suppose 5^^ = and the order

2.0 5 66

is obeyed; this is now effectively an unconditional Jump,
later, by such orders as the following, for example.

but it sets 5j^ = 1 and this can be sensed

35 5 03

3.0 5 66

Here there will be no jump if 5p was 0. Care should be taken lest such orders set OVR however.
We have described in Sec. 5.5 how pseudo order-pairs may be used for setting modifiers and
counters. Sometimes a relative modifier may be used: this is especially useful if the part of the
programme being considered starts beyond the first part of the store, and programme transfer -orders
must be modified. For example, the following might be the start of a subroutine.

. pseudo order-pair = (0+. 0, 0)

set 2g = 0+
read B3+ into (71

0.0

0+ - 00 0.

3 [T]72

A subroutine written like this may be placed anywhere in the main store.

Sometimes only a modifier is required and we are not interested in the value of the associated
counter; we may be able to use a dummy a-order to give us this. For example, the following order-pair
might appear in a programme.

0.6

j 50 00

I 5.3 2 10

Dummy order

The modifier 0. e^j^ is here 400.0 (400 = 8 x 50), and the a-order has no effect (except to waste a very
small amount of time). This word could be copied into an accumulator to give a modifier of 400.0; such
an order-pair should be marked by a vertical broken line on the left to indicate that it is also used
as a constant; this is to warn those who might wish to alter the routine.

We can sometimes use an actual order-pair to provide a modifier, particularly when only the
position part matters (see Sec. 5.5), or when we are modifying only Jump orders. For example, consider
the following word:

|2.5+ 3 61

2.0

1 ® 6 40

L__

The modifier 2. 0^ can be worked out from a knowledge of how the parts of an order are represented (see
Sec. 3.12); the block part 2. O^g is formed from the N- and A:-digits of the a-order and is here equal to
253 in octal, or {(2 x8)+5}x8+3= 171 in decimal; the position part, 2. Op, is 6; so the whole
modifier 2.0nf is 171.6. Note that such an order-pair may be set either by an OO-order or, alternatively,
by an 02-order. If the above order-pair is set by an 02-order the resulting modifier is 1023.7 -
171.6 = 852. 1. Such devices should be used with caution. Note that we can remove an unwanted counter
from a modifier by a pair of orders such as the following.

35 5 02

2.0 5 05

- 118

MODIFICATION

5.10

An ingenious application of this technique^, which is definitely a trick, is the self- modified
link. This is a link which, when obeyed, is simultaneously in an ordinary register and in an accumulator
(usually XI); consider, for example, the following sequence of orders.

0.0

0.1 1 00

100 [U 72 1

3.1 60

I?903 to f/g

The order-pair in 0. 1 is in XI when it is obeyed in 0. 1; the block part of the modifier Ig is equal
to (100 X 8) + 3 = 803 so that the modified 72-order actually transfers J5903 = 803 + 100, Such an
order-pair can be used to return from a subroutine to a programme held beyond the first part of the
store, provided of course that XI is untouched by the subroutine. Only certain blocks can be
transferred by such a link; for, consider a general self-modified link as follows.

B \U]'72 1
U.P 60

The block part of the modifier (which is all that is relevant) is 8B + U so that the block transferred
is 9B + U. For example only blocks whose numbers are divisible by 9 may be transferred to C/0.

The above tricks are based on using an order as a modifier. Sometimes we can use a part of a
number as an order; the only really useful trick consists in using the "a-order" of a word normally
used as a small negative integer; this is a 77-stop.

There is a range of tricks based on using as a modifier a word which is numerically a small
integer. The modifier is zero if the integer is non-negative, and is 1023.7 if the integer is
negative.'' For all but the arithmetical orders a modifier of 1023.7 can often be treated as having
the value -1. Consider, for example, the order

3+[o]72

If C(6) is a small non-negative integer, this order reads B3+ into t/O; if, however, the integer is
negative then the above order will read B2+. In the same circumstances the order

0.5+ 60 6

will jump to 0.5+ if C(6) ^ 0, or to 0.4+ if C(6) < 0.
in one order, such as

We can sometimes in this way combine two tests

2.0 3 63 6

which will jump to 2.0 if C(6) ^ and C(3) < 0, or to 1.7 if C(6) < and C(3) < 0. and will not cause
a jump if C(3) ^ 0; assuming C(6) is not a large integer. It is sometimes useful to test OVR
by an order of this kind.

Note that if we use this kind of trick with any of the arithmetical orders then it is only the
position part of the modifier which is used; this has the value 7 if the small integer is negative.
The address written in the order is never decreased by modification (unless it refers to Ul). For
example, the order

5.6 4 01 6

will add C(5.6) into Xi if C(6) > 0; but if C(6) is a small negative integer then the operand is
C(6. 5), which is zero (see Sec. 3.9). Similarly, the order

5.7 4 05 4

will collate C(4) and C(5.7) if C(4) is a small non-negative integer; but the order will clear ..Y4
if it contains a small negative integer. Another useful order of this kind is the following

7.5 4 12 4

If X'i contains a small non-negative integer this order will have no effect; if however C(4) is negative
then 4p = 7 and the modification changes the effective address into 7.5+7-8.4 and this is equivalent
to 4 because of the way the A^-address is represented (see Sec. 3.12) and the fact that only the 7 least-
significant digits in the modified A'-address are effective (see Sec. 5.8). If, therefore, C(4) is a
small negative integer, the above order has the same effect as the order

4 4 12

t Due to Mr. D.G. Owen.
t+ In fact 6^ = if < C(6) < 2^5, and 6jj/ = 1023.7 if -2^^ ^ t;(6)

- 119

5.10 MODIFICATION

and C(4) is changed in sign. The net result is that the order written above replaces C(4) by its
modulus. Similarly the order

7 12 6

can be used to change the sign of C(7) if C(G) is negative.

We call now explain the trick we referred to in Sec. 3.8 for correcting the sign-bit in XI after
the multiplication of two possibly negative integers. Suppose that we have two such integers in 5.0
and 5.1, then the orders

5.0 7 00 first integer to XI

5.1 7 20 multiply by second integer
4.0 7 10 product to 4.0

will put the product correctly into 4.0 provided it is non-negative and less than 2^^ (see Sec. 3.1).
If the product is negative we have to insert a 1 in the sign position in XI before using the product;
we gave in Sec. 3.8 the following orders for doing this.

6 62

32 7 01

►

We can now see that these two orders can be replaced by the following single order.

25 7 01 6

If the product is non-negative 6p is zero and the content of special register 25 is added into XT. This
does not change C(7) since C(25) is zero (see Sec. 2.9). If the product is negative 6p is 7 so that
the effective address in the order is 25 + 7 = 32 and the 1 is inserted in the sign position. We can
usefully apply another trick mentioned earlier in this section to detect overflow, which in this con-
text means that the product is not single- length. If the product is single-length then C(6) as an
integer is either or -1 and therefore XG will be cleared by an arithmetical halving. The complete
sequence may therefore be written as follows.

5.0 7 00 first integer to XI

5.1 7 20 multiply by second integer
25 7 01 6 correct sign in XI

(T) 6 51
Cf 6 61

4.0 7 10 corrected product to 4.0

■ stop if product Is not single-length

The overflow test may, of course, be omitted or adapted if appropriate.

The orders with functions 05, 06, 15, 16, 45 and 46 may often be used to provide trick effects.

The reader will have noted that none of the tricks described in this Section does anything that
could not be done by more conventional methods. All that can be said for them is that they save
orders and a little computer time, at the expense of increasing programming time and the difficulty of
understanding or altering the programme later. They are sometimes useful when a small mistake has been
discovered in a programme and an extra order or two are needed to put it right (this is quite a
common situation): it may then be possible to make enough room by using a trick in an adjoining
section of the programme. ^ To make these small corrections easy to put in it is recommended that
those parts of a programme where timing and compactness are not important should be written in a most
straightforward way, and that a few dummy orders should be left in each block. Quite often there
will be two good ways of doing a small piece of calculation, but one of them might be slightly quicker
and take one or- two more orders. If this way is chosen, then we can always substitute the other one at
a later date to make room for a correction, should this be necessary.

There Is a well known "theorem" to the effect that any programme can be shortened by one order!

- 120

Chapter 6

Input and Output

This chapter describes the input and output devices and the tapcediting equipment for handling
punched paper tape. The rules are also given for preparing programme tapes. There is in addition a
detailed description of the various controls, hand' switches and monitors and how to use them.

6. 1 Punched paper tape

The equipment fitted in a basic Pegasus installation so that orders, numbers and other information
can be inserted into the computer and extracted from it uses standard five-hole (or five-channel) punched
paper tape. This tape is about 11/16 in. wide (1.746 cm.) and carries a row of small holes punched ten
per inch along its length; these are the sprocket holes, which are used to move and position the tape.
Each sprocket hole defines the position of a tape character, which is made up of a maximum of five holes
(each slightly larger than a sprocket hole), punched in a row across the tape and in line with the
sprocket hole. Three of the five channels are on one side of the spi'ocket hole and two are on the other.

The input equipment is normally two photo-electric tape-readers (Perranti type TR5) to read the
input tapes into the computer (plate 3). these read up to 300 characters per second. The output
equipment is normally a single output punch (Teletype) which can punch up to 60 characters per second
(plate 4). This output tape is usually fed immediately into a Creed interpreter, consisting of an
automatic tape transmitter (a kind of tape reader for reading the tape) connected to a page teleprinter
(Creed model 75), which we shall often refer to as a printer; + this Interpreter prints 10 characters
per second. The output punch is six times as fast as the interpreter and it usually operates in bursts.
If a programme causes a great deal of output then a loop of tape will form between the output punch and
the automatic transmitter of the interpreter. If the tape should become taut a taut-tape device on
the tape transmitter stops the transmitter so that the tape is not torn.

Normally the printer will print a single character (e. g. a decimal digit or a letter) each time the
automatic transmitter reads a tape character. With five-channel tape there are just 2^ = 32 different
combinations of holes, each of which is a distinct tape character. Since we want to be able to use more
than 32 different printed characters it is arranged that most of the tape characters correspond to two
printed characters each. The character which is actually printed depends on whether the printer is in
figure shift or in letter shift. There are two special characters which determine which shift ip to be
used for subsequent printing. When the printer receives the figure shift character (usually abbreviated
to the Greek letter (phi), or sometimes to PS) then it will print all subsequent characters in figure
shift until it receives a letter shift character (usually abbreviated to the Greek letter K (lambda), or
sometimes LS), when it will start printing in letter shift. The characters and k do not themselves
cause any printing. The two shifts resemble in some ways the upper and lower case shifts obtainable on
a typewriter. We assumed in Sec. 4. 1, where we described the output of numbers, that the printer was in
figure shift; this is the normal or standard shift.

The complete tape code used with Pegasus is given in Table 6. 1. The first column shows the tape
characters and the last two columns the corresponding printed characters in figure shift and letter
shift. The characters (figure shift), k (letter shift), full stop and Erase are common to both shifts.
The other 28 tape characters each correspond to two printed characters. The two columns headed 7V(17, 16)
are explained below. The characters CR, IF and Sp are the layout characters described in Sec. 4. 1; they
are available only in figure shift. The Erase character has holes in all five positions on the tape;
this character is printed as a star (Jt)in either shift and is usually referred to as Er. Any tape
character can be converted into Er by punching extra holes in it; for this reason it can be used for
correcting tape-punching errors and we usually arrange to ignore any Er characters on the tape. Notice
that the character (figure shift) is the blank character; it has only a sprocket hole. A number of
successive s is usually called a length of blank tape, though it is not strictly blank because of the
sprocket holes. All our tapes will start with a leader of about 6 or 8 inches of blank tape to
facilitate their insertion into the tape readers and automatic transmitters. This automatically ensures
that the first character after the blank tape will be read in figure shift (the normal shift). The
output punch and the units of tape-editing equipment for preparing tape are each fitted with a run-out
button, which causes blank tape to be punched as long as it is held down.

The tape code has the property that the most used characters (in figure shift) have each an odd
number of holes. These odd-parity characters are the decimal digits (0 to 9), the signs (+ and -), full
stop (©), LP, Sp and Er. The reason for choosing a code with this property is that it makes a certain
amount of checking possible. The majority of errors due to faulty operation of the input or output

t Two telegraphy terms: an automatic tape transmitter is a device which reads punched paper tape
and transmits electrical signals to a teleprinter, and a page teleprinter is a teleprinter which
produces a print-out in page form - not in the strip form seen on Post Office telegrams.

- 121

6.2

INPUT AND OUTPUT

TAPE

PRINTER

PIGS.

LETS,

FIG. SHIFT (c6)

•

•<

(

•

)

•

••

L F

«•

••

•

9 9

••

e»

••

«•

LET. SHIFT (k)

«••

•••

C R

•••

Erase iZ)
1

Table 6. 1 The tape code.

devices or the tape-editing equipment result in the gain or loss of a single hole when punching tape,
or the misreading of a single hole when reading tape. This kind of faulty operation will always
convert an odd-parity character into an even-parity character, but cannot^ for example, convert one
decimal digit into another. A single fault will usually produce an obviously wrong result; for
example, a missing hole could turn a 3 into an asterisk, or a 9 into a multiplication sign. This
checking is an example of a parity check, which finds many applications in computers (see also Sec. 6.7),

The design philosophy in Pegasus assumes that the majority of errors are due to the input and
output equipment, the tape-editing equipment, and the stores. These devices are all subject to a
parity check (see Sec. 6.7). Errors due to human frailty can often be detected or prevented by
careful programming.

It should be noted that many different tape codes are in current use with different computers and
other devices. The code described above is the standard Pegasus/Mercury tape code.

6. 2 Output

The two special registers whose addresses are 16 and 17 are used for input and output. We have
already described In Sec. 4.1 how most of the useful characters can be punched by sending a small
integer (the value of the character) to register 16. In fact we can also punch a character in the
same output tape by sending the integer to register 17, but the resulting character will in general
be different.

122

INPUT AND OUTPUT 6.2

When we send an integer (less than 32) to register 17 it can be thought of as going directly to
the punch; Its binary digits are punched as a hole for a 1 and no hole for a 0. For example, the
orders

(j) 2 40

17 2 10

will cause the tape character for an asterisk to be punched (if this character is read in letter shift
it corresponds to the letter C). This character has the two right-hand (or "least-significant") holes
punched, it may be written as 000.11 if we write a 1 for each hole, a for the absence of a hole, and
a point for the sprocket hole. If, on the other hand, we send an integer (less than 32) to register
16, then it undergoes a conversion before being punched. If the integer is 15 or less, then it is
adjusted (if necessary) so that the tape character punched has an odd number of holes - this is done by
inserting the left-hand (or "most-significant") hole when required. For example, the orders

2 40

16 2 10

will cause the tape character for 3 to be punched. Since 3 ^ 15 this character has odd parity; it may
be written 100.11, and could alternatively have been punched by sending 19 to register 17. A similar
process is applied to integers greater than 15 to ensure that the resulting tape character has an even
number of holes. Thus the orders

19) 2 40

16 2 10

will cause punching of the tape character for asterisk. Since 19 > 15 this character has even parity.
The main purpose of register 16 and its conversion equipment on output is to make it easy to punch
the decimal digits (see Sec. 4.1).

It will be seen that a tape character has two values, one via register 16 and the other via 17;
these two values are given in Table 6.1 in the columns headed /V (17, 16). Generally we use register
16 since the decimal digits and a few other commonly used characters have convenient small values, as
given in Table 6. 2.

Printed
Character
(in figure
shift)

Value
via 16

12 3 4 5 6 7 8 9+ - . LF Sp Er CR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 30

Table 6.2 Values of some commonly used figure shift
characters via register 16.

Register 17 is usually more convenient than 16 for the output of letters of the alphabet, because
the value of the n-th letter is simply n (i.e. /! has the value 1, B is 2, C is 3, and so on). Note,
however, the useful order

17 10

which causes to be punched. If this order is obeyed repeatedly we get a length of blank tape.

Whenever a word is sent to the output punch, whether via 16 or 17, it is the five least-significant
digits only which determine the character to be punched; these are digits 34 to 38. Digits to 33
have no effect on the punch. The punch can be operated by any order of group 1 whose Ar- address is 16
or 17 (it is occasionally useful to use a 12-order in some tricks); such an order is called an
output order.

The normal output punch operates at a maximum speed of 60 characters per second, so that it takes
about 17 milliseconds (or about 133 word-times) to punch a single character. This time is long
compared with the time required by the computer to obey a single order. When a character has been
sent to the punch by an output order (via 16 or 17) the computer does not wait for the punch mechanism
to operate but carries on and obeys orders in the usual way. In other words, the punch operates
independently or autonomously while the computer is obeying further orders. Should another output order
be encountered before the punch mechanism has finished, then the computer is made to wait. In fact
during the 17 milliseconds that the punch is operating it sends a continuous busy signal to the computer;
this signal holds up the computer if it is about to obey a further output instruction. In Sec. 4. 1 we
disregarded all timing considerations when dealing with the output punch; this cannot result in incorrect
operation. There is a paper tape light on the control panel (Plate 11) which is lit while the punch (or
the tape-reader) is operating, t

^ On Pegasus 1,' the output busy signal holds up the computer if it refers to registers 16 or 17 on an
input or output instruction. However, on Pegasus 2, each of the standard input and output channels
has its own individual "busy" line.

On the control panel of Pegasus 1 are two "busy" lights labelled "input busy" and "output busy" (although
on Pegasus 1 with magnetic tape equipment the "output busy" light is replaced by "magnetic tape busy").

- 123 -

6.3 INPUT AND OUTPUT

When the tape has been punched it normally passes straight to the interpreter for printing at 10
characters per second. The interpreter is for convenience mounted on the control desk of the computer
(Plate 4), tmt can be considered as entirely separate from it since what happens to the output tape when
it has been punched does not in any way affect the computer.

Apart from occasional punching of layout characters (CR, LP and Sp), most of the output require-
ments of programmes are met by subroutines. We discussed in Sec. 4. 1 the way in which these subroutines
can be prepared.

6. 3 Input

Input can be thought of as the converse of output. There are two tape-readers, referred to as the
main and second tape-readers, which can be used to sense photo-electrically the holes punched in the two
input tapes and to move the tapes. The external-conditioning order (function 74 - see Sec. 3. 10) selects
which of the two tape-readers is to be used. We shall simply refer to "the tape-reader" when we mean
the selected tape-reader, and to "the input tape" when we mean the tape in this reader. We can read a
single character from the input tape by taking a word from registers 16 or 17. For example, the order

16 2 00

is called an input order or read order; it will place in X2 the value of the character in the reading
position in the tape-reader. The value via 16 (as described in the previous Section) is the one used.
As soon as the tape has been read the tape-reader automatically moves the tape forward so that the next
input order reads the next character on the tape. For example, if the character in the reading position
is 3 and the next character is Sp, then the orders

16 2 00

16 3 00

will place 3 in X2 and 14 in J3.

Suppose, for example, we have to read three characters representing decimal digits from the tape
and build up the corresponding binary number in XI. For example, if the characters are 2, 4 and 9 then
249 is to be put into ^7; we can use the following relationship.

(2 X 10 + 4) X 10 + 9 = 249.

The following sequence of orders will carry out this operation.

@ 5 40 10 to X5

16 7 00 first digit to XI (e.g. 2)

5 7 20 multiply by 10 (e.g. 20)

16 7 01 add next digit (e.g. 24)

5 7 20 multiply by 10 (e.g. 240)

16 7 01 add last digit (e.g. 249)

Note how these orders effect the decimal to binary conversion. They do this mainly because the tape
character for any decimal digit has a value equal to that digit when read via register 16.

The process of input via 16 is the exact converse of output via 16. Special checking equipment
is used which converts the tape character into an integer of value 15 or less if it has an odd number
of holes, or into an integer greater than 15 if it has an even number of holes. This greatly
facilitates the detection of punching errors, tape-reader malfunctioning, or the blocking of holes by
pieces of fluff.

A sequence of orders like the above would not be used in practice, because it does not include any
checks on the punching and because the preparation of a tape with many numbers would be very laborious
and difficult to check. The sequence could be described as a primitive input routine, since the way in
which the tape has to be prepared is rigorously prescribed. For example, there must always be just 3
decimal digits on the tape, with no other characters (left-hand zeros must be punched as such).

Generally speaking input, even more than output, is a process which is best left to a subroutine.
Certain punching rules will have to be followed when preparing the tape for any particular subroutine;
if possible the subroutine should be written in such a way that its punching rules are convenient and
natural. What seems natural and simple to the person preparing the tape may be difficult and complica-
ted when expressed in the form of precise rules. For this reason many input subroutines are complicated
and difficult to write; they must give a certain amount of latitude to the operator of the perforator.
For example, it is highly desirable to ignore Er (the Erase character) wherever it occurs, A certain
amount of checking is useful so that spurious characters among the significant ones cannot give mis-
leading results.

As a simple illustration of some of the ideas involved let us construct a subroutine to read in an
unsigned integer from the input tape, convert it to binary and leave it in XT. The subroutine will read
in one by one the characters representing the decimal digits of the number until it encounters a Sp,
which is taken as terminating the number; Er will be ignored. As each character is read in it has to
be tested because it may be any of the following:

(a) a decimal digit,

(b) Sp.

(d) some other character.

- 124

INPUT AND OUTPUT

6.3

If it Is (d) we shall have a loop stop to indicate a fault or (more likely) a punching error. If the
character is Er we must return to read the next character. If it is Sp we must obey the link. On
entry, the subroutine clears X? then, whenever a decimal digit is read, C(7) will be multiplied by 10 and
the new digit added to it. The following is a possible subroutine.

0.0

Oil

7 00

0.3

5 41

6 40

7 20

7 01

5 00

5 43

0.1

5 63

5 43

0.3

5 60

5 41

0.6

5 61

0.7

1 10

( )

clear Xl to receive integer
enter main cycle
C(5) = t (a digit)

multiply partial result by 10

add new digit

read tape, C(5) = t, say

t - 10

jump if t is a digit (t ^ 9)

t - 15

jump if t is Er (t = 15)

t - 14 (= if t is Sp)

stop if not Sp

plant and obey lirk from XI

This subroutine would not normally be acceptable, for a variety of reasons. If, for example, the
first character read is Sp then the link is immediately obeyed; as a rule It would be desirable to
ignore Sp before the first digit is encountered, but to treat it as a terminating character afterwards.
Again, we would usually want to treat CR LP (i.e. CR immediately followed by LP) in the same way as Sp,
so that we could have numbers on several lines. Often we should want to handle signed integers and we
must then arrange to accept the sign characters (of values 10 and 11) before (but not after) the digits
and to change the sign of the number, if necessary, at the end just before obeying the link. It would
probably also be desirable to ignore blank tape under certain circumstances. When an input subroutine
is written to meet all these and other requirements it becomes quite complicated; each character is
subjected to a whole series of tests and checks before any other action is taken on it. Subroutines
like these are to be found in the Library; they are easy to use and it Is easy to prepare tapes for
them. Generally speaking, each subroutine will be designed for some particular kind of number -
integers, fractions, mixed numbers, sterling, and so on; and different verions of certain subroutines
are available.

Input via register 17 is the converse of output via 17 except that the integer resulting from reading
the tape, appears in the modifier position. For example, if the character in the reading position is 3
then the input order

17 2 00

will put 19 into 2j|^ and clear the rest of X2 (see the tape code in Table 6.1). Input via 17 is not often
used but is very convenient when some sort of code conversion is to be applied to the characters as they
are read in; this is why the modifier position is used.

A ta,pe-reader of the type normally fitted operates at a maximum speed of about 300 chara(3ters per
second, so that it takes about 33^ milliseconds (or about 27 word- times) to move the tape from one
character to the next. When a character has been read by an input order (via 16 or 17) the tape-reader
immediately starts to move the tape to the next sprocket hole (or character); it does this quite
independently of what orders the computer may be obeying - another example of autonomous operation.
Should another input order be encountered before the tape has finished moving to the next character,
then the computer is made to wait and the tape is not brought to rest, the character being read as the
tape moves by. In fact during the dVz milliseconds that the tape is being moved the tape-reader sends
a busy signal to the computer; this signal holds up the computer if it is about to obey a further
input instruction. t if all timing is disregarded incorrect operation cannot occur. The paper tape
light on the control panel is turned on by the busy signal from the tape-reader, tt We attempt to write

t On Pegasus 1, the input busy signal holds up the computer if it refers to register 16 or 17. Thus
the busy signal sent by either the tape-reader or the punch inhibits all reference to registers 16
and 17. This means that the computer will be made to wait if it encounters an input order while the
output punch is operating, and vice-versa. The fact that the busy signals are not distinguished
within the computer does not often matter.

tt On Pegasus 1 a busy signal from the tape-reader lights an "input busy light".

125

6.4

INPUT AND OUTPUT

input subroutines so that the computer takes rather less than 3^3 milliseconds between input orders; the input
tape is then kept running at full speed and the input busy light Is on continuously while the tape is being read
in. Note that the tape will always stop in time if there is a long interval between one input order and the next.

A code conversion can often be achieved by a table look-up operation (see Sec. 5.8). There is an
interesting processt for doing this which is well suited to the kind of conversion often required on
input. Here we have a 5-bit integer t (0 ^ t ^ 31) which will normally be the value of a tape
character read via register 17. We have to convert this integer into another integer u by looking up
the t-th entry in a 32-entry table. Often we shall not want to use a whole word for each entry and we
can therefore pack several entries into one word. If we can pack four entries into each word the whole
32-entry table can be held in the 8 words of a single block. Consider the following sequence of orders,
where the table is supposed to be in f/4.

17 2 00

t to 2.

4.0 6 00 2 extract fe-th word of table

(o) 6 52 2 shift up t places

The first order simply reads the tape in the "direct" way and puts t into 2^. The next order is an
arithmetical order and will therefore be modified by the last 3 bits of t only (let us call this k; it
is the remainder when t is divided by 8). The last order is modified by the whole of t since it is an
order of group 5. For example if t = 20 then the remainder fe is 4 and the above orders will pick out
the word in 4.4, copy it into J6 and shift it up 20 places; if t = 28 the same word will be put into
X6 but it will be shifted up 28 places. The 8 most-significant binary digits in X6 constitute u, the
result of the table look-up. The rest of C(6) can be discarded if required by shifting or collating.
These 8 bits can be arbitrarily assigned for each of the 32 entries. The whereabouts of each entry
are given in Pig. 6.1; each numbered space represents an 8-bit entry, the numbers being the value of t
needed to extract the entry in the above way. The shaded parts are not accessible.

4.0

4. 1

4.2

4.3

4.4

4.5

4.6

4.7

Pig. 6.1 A packed 32-entry table.

This block would normally be written on the programme sheet as eight pseudo order-pairs (see
A Sec. 3.12).

6.4 The tape-editing equipment

The equipment for the manual preparation, editing and printing out of the punched paper tape is
called tape-editing equipment. The following facilities are provided.

(a) Punching tapes manually from a keyboard, with or without simultaneous printing.

(b) Printing out tapes.

(c) Copying tapes with simultaneous printing and, if required, the insertion of corrections from a
keyboard or another tape.

(d) Comparison of two tapes to detect discrepancies due to punching errors, with or without the
simultaneous preparation of a print-out and a third "perf ect" tape free from errors.

t Due to Mr. D. G. Owen.

126

INPUT AND OUTPUT 6.4

The simplest item of equipment is the keyboard perforator (Plate 9) which consists simply of a
keyboard which operates a tape-perforator. It can be used only for the manual preparation of tapes
without printing, and is consequently used only for very short tapes. There is usually a keyboard
perforator near the computer.

An important unit In all installations is the page teleprinter (Creed Model 75) with keyboard and
reperforating attachment. It can be used by itself for the manual punching of tapes with simultaneous
printing, or it can be connected to other devices to provide full editing facilities. It provides
the most convenient means for preparing tapes manually because a printed record (a print-out) Is
obtained while the tape is being punched. The keyboard of this instrument (Plate 10)' resembles that of
a typewriter except that the keys for the decimal digits (and certain other characters) have been
grouped together at the right. This grouping makes it possible to prepare most programme and data
tapes with one hand only. The run'out key is at the top right corner of the keyboard; if it is
depressed, continuous punching of any character may be obtained by also depressing the key of the
character required. The run-out will cease when either key is released. + The operation of any other
key causes a single tape character to be simultaneously punched and printed, except that a few characters
produce no actual printing (these are cb, k, CR, LP, Sp). Figure shift (0) is punched by pressing the
bar labelled PIGS/BLANK and letter shift (k) is punched by operating the key on the left labelled LTRS
(for "letters"). The keys are engraved with the printed characters, and there are consequently two
keys for each tape character (with a few exceptions). The keys for the characters available only in letter
shift (chiefly the letters A, B, ..., Z) are locked when the teleprinter is in figure shift, and vice-
versa. Thus the key engraved B is free only in letter shift and the key engraved 2 is free only in
figure shift; these keys cause the same tape character to be punched. The keyboard perforator has a
similar keyboard.

Most punching errors aije noticed immediately they have been made; the operator can back-space
the tape and over-punch incorrect characters with Erase (Er, which has all 5 holes punched); the
over-punching (but not the back-spacing) is printed out simultaneously. The keyboard perforator has a
similar facility.

The printing is on a continuous roll of paper wide enough for 69 characters. There are 10
characters to the inch horizontally, and there are 6 lines to the inch vertically. The tape is produced
by the reperforating attachment and comes out from the side of the teleprinter; the action of this
attachment can be suppressed if desired, when the teleprinter will print without punching.

The simplified tape- editing equipment (Plate 8) consists of a teleprinter of the kind just
described connected to an automatic transmitter (a kind of mechanical tape-reader) and a small control
box; the equipment is sometimes called a reproducer. The automatic transmitter can be used to read a
tape and operate the teleprinter, which then prints what is on the tape and punches a new copy of it
(unless punching is suppressed). At any time the automatic transmitter can be stopped and characters
inserted from the keyboard. When the teleprinter is operated from the automatic transmitter the
printing is in black; when the keyboard is used the printing is in red (the latter is usually the
result of manually inserted corrections). The tape in the automatic transmitter can be inched forward
character by character by operating a key on the control box; the characters inched over in this way
can be copied or ignored as desired.

This set of equipment is adequate for the preparation of most tapes. The print-out which is
obtained makes it easy to prepare and edit a tape since it gives a visible record of operations. The
usual procedure for preparing a programme tape is as follows. Pirstly, the tape is punched by an
operator directly from the programme sheets (the punching rules are described in Sec. 6.5). Secondly,
the print-out obtained during the punching is proof-read against the original programme sheets and any
errors are marked on the print-out. Lastly, the tape is reproduced with the aid of the automatic
transmitter, and the resulting print-out is watched. Just before a punching error is reached the
copying is stopped and the tape inched up to the last correct character; the incorrectly punched
characters are then inched over one by one without copying and corrections are put in on the keyboard.
This procedure is repeated for each error. When the end of the tape is reached the new print-out will
have all corrections in red and can be quickly checked. Note that one can prepare and correct a tape
in this way without learning the tape code; nearly everything can be done by watching the print-out.

The simplified tape-editing equipment can also be used for joining up a number of short tapes into
a longer one by copying. As a rule this is not recommended since it can take a long time and errors
may be inserted during the copying. It is better to splice the several tapes together, as described
below.

The full set of tape- editing equipment (Plate 7) provides all the facilities of the simplified set
and can in addition be used for comparing two tapes, and correcting errors discovered as a result. It
is made up of the following items of equipment.

(a) a page teleprinter with keyboard and reperforating attachment, as described above,

(b) an automatic transmitter (as in the simplified set),

(c) a triple- head tape-reader or comparator (two of the heads are used when comparing tapes automati-
cally, while the third can be connected to the teleprinter for copying or printing out tapes),

(d) a control unit,

(e) an independent keyboard perforator.

The equipment also includes a power unit. The keyboard perforator is an entirely independent item but
the other units mentioned above are all connected together.

The method described above for the preparation of programme tapes is not suitable as a rule for
data tapes. Programme tapes are often prepared by the programmer himself, who can easily read and

t On the previous model of -the teleprinter (Creed Model 54), it was only possible to run-out the blank
character (i.e. 4> or figure shift).

127

6.5 INPUT AND OUTPUT

understand the programme sheets and who will be on the watch for slips and other minor errors, many of
which he will probably detect and put right while he is punching the tape. Programme tapes are also
fairly short and easy to punch and check. The punching usually comes as a welcome relaxation to the
hard-pressed programmer! Any errors which slip through the check are usually picked up quite quickly
when the tape is fed into the computer. It is quite otherwise with data tapes. These are nearly always
produced from lists of figures with no apparent meaning and may be very long. It is also difficult
to detect errors by proof —reading and the consequences of the errors may be serious. It is therefore
recommended that data tapes should be prepared twice, preferably by different people using different
equipment but working from the same data sheets. The two tapes should then be identical but, in all
probability, each will contain a number of errors (perhaps one error in every 300 to 2000 characters).
These two tapes can be automatically compared by the comparator section of the tape-editing equipment,
this is normally connected to the teleprinter which produces a print-out and a third tape (the so-called
perfect tape). The copiparator reads both of the hand-punched tapes, character by character; as long
as they agree the third tape is punched. The device stops as soon as there is disagreement and the
print-out can then be inspected to determine the correct next character, which will usually be on one
of the two input tapes. This tape can then be chosen as the "master" and copied on to the final correct
tape. Alternatively a correction can be put in from the keyboard of the teleprinter, or from the
automatic transmitter. The comparator can also be disconnected from the teleprinter and used simply to
compare two tapes without producing a third; this facility is mainly used for comparing a copy of a
tape with the original.

The full set of tape- editing equipment also includes an automatic carriage-return and line- feed
facility to return the carriage of the teleprinter after a preset number of characters have been
printed across each line.

The interpreter mounted on the desk of the computer (Plate 4) is conveniently described here,
although it cannot be used for editing tapes. It resembles a simplified set of tape-editing equipment
in that it includes an automatic transmitter and a page teleprinter. However, there is no control box
and the teleprinter has no keyboard and no reperforating attachment.

The comparator works at 4.4 characters per second when comparing and producing. The model 75
printer works at 10 characters per second and the model 54 printer at 6.6 characters per second.

In addition to the major items of equipment described above there are a number of tape- handling
accessories.

The motor-driven tape-spooler (Type Al) can be seen near the second tape-reader in Plate 3. It is
primarily intended for feeding a long tape into a high-speed tape-reader or for winding up a tape
issuing from it. The hand-operated tape-spooler (Type A3 - see Plate 6) is for spooling tapes out of
bins; there are usually one or two near the computer and the tape-editing installation. Special tape
boxes (type A4 - see Plate 5) are used to hold tapes at all stages of their preparation and use; each
box has a core (type A5) which rotates on a pin in the box. These boxes have a slot in the side to
allow the tape to be fed out or spooled up inside the box. The spooling up can be done either with
the hand-operated spooler or by a spooling knob (type A6). Tapes are usually fed into the tape-readers
on the computer from a plastic trough (Type A7), one of which can be seen in Plate 3; if desired the
tape box can be put into one of these troughs and the tape pulled out through the slot in the box. A
number of tape bins (Type A8) mounted on castors are normally used to receive the tape coming out of the
tape-readers and punch on the computer, and also from the tape-editing equipment. Small cardboard boxes
are used to hold short tapes, chiefly programme tapes. The tape-correcting punch with splicing
facilities or unipunch is a useful device (see Plate 13); it has two functions, as follows;

(a) It can be used to insert extra holes or a few extra characters in a tape.

(b) It assists the splicing or joining of tape.

A partly spliced tape is shown In Plate 13. The two tapes to be joined are placed in the device, which
holds them firmly. The two ends are then cut straight and are covered with opaque adhesive tape (PVC
tape) to make a butt joint (end to end); the tapes are then taken out of the device and the adhesive
tape is wrapped round the underside of the two punched tapes. This kind of joint is normally made only
where there are lengths of blank tape; it covers over the sprocket holes but this does not affect the
operation of the high-speed tape-readers on the computer. Such a join cannot, however, be passed through
the tape-editing equipment. In. general this method of joining is preferable to copying with the tape-
editing equipment; it is quick and easy to follow.

By using all these various items of equipment it is possible to prepare and edit punched tapes for
the computer. Great care should be taken that the tapes are correct before they are fed to the computer
and, if at all possible, numerous short sections of tape should be spliced into a long tape in order to
save computer time, and prevent errors.

6. 5 The preparation of programme tapes

We shall now describe how a programme tape is prepared. We shall assume that the punching is done
on a page teleprinter (with keyboard and reperforator attachment) as described in the last Section. We
therefore get a print-out while the tape is being punched. If the tape is instead punched on a keyboard
perforator then the only difference is that no print-out is obtained; the actual punching procedure
is unchanged. The description which follows is given in the form of rules, which should normally be
adhered to. Sometimes departures from the rules are allowed, but these will not usually be mentioned
and the beginner is advised to follow the rules carefully. In general it will be found that the punching
rules are natural and easy to follow and will result in an acceptable print-out for proof-reading. We
assume that the complete programme or subroutine is all written out, complete with directives, ready for
punching. A few additional rules are needed to cover certain punchings required by Assembly, and some
special directives; these will be described later. We assume that the programme is to be read in by
the Initial Orders (see Sec. 4.3).

A complete tape is made up chiefly of the following;

(a) Lengths of blank tape,

(b) Order-pairs,

(d) Directives.

- 128 -

INPUT AND OUTPUT 6.5

When the tape is read in by the Initial Orders the blank tape is, by and large, ignored (i.e. it has no
effect); the order-pairs and numbers are placed in consecutive storage locations as determined by the
Transfer Address, and the directives control the operation of the Initial Orders and are not stored.
These four kinds of punching are distinguished from one another by their first characters, apart from a
few which have no effect, like Er; these first characters are as follows.

(a) Figure shift (0) introduces blank tape,

(b) A decimal digit (0 to 9) introduces an order-pair,

(d) Letter shift (A.) introduces a directive.

The reason for treating blank tape in a special way is primarily to detect faults in the tape- reader
or the tape-editing equipment. Pull use is made of a check rendered possible by the special features of
the tape code and register 16 (see Sec. 6.3). Most of the characters used for punching are the odd-
parity characters (see Sec. 6.1), viz.

0123456789 + -. LPSpEr

which have each an odd number of holes in the tape. An error of a single hole in reading or punching
cannot turn any one of these 16 odd-parity characters into another one. Blank tape (cp) is not an odd-
parity character, and in fact a single error could cause confusion between and any of the characters
0, 1, 2, 4 or 8. For this reason it is not desirable to accept blank tape among the digits of an order-
pair or number, and some preliminary search is required. In fact if is encountered in the middle of
an order-pair or a number the Initial Orders will encounter a loop stop.

The following are useful rules about blank tape.
Al, All tapes should start with a leader of at least 6 inches (15 cm.) of blank tape (obtained by

pressing the run-out key); the name of the tape, the date and the programmer' s (or punch operator ' s)
name or initials should be written on this leader in ink or pencil. This leader facilitates the
insertion of the tape into tape- reader s or tape- editing equipment.

A2, About two inches (5 cm.) of blank tape should be punched at any natural breaks in the information

being punched; for example, after any of the following:

(a) each block of the programme,

(b) each directive (or warning' character sequence) ,

These short lengths of blank tape greatly facilitate splicing the tape and identifying portions of it.

A3. The characters CB and LF should always be punched before anything else after any blank tape.

Thus each directive or each block of programmeis normally preceded by blank tape CR LF.

Ah. If there are several routines or long lists of numbers on the same tape it is advisable to punch

about 6 inches (15 cm.) of blank tape between them, and to write here the name of the immediately

following routine or list of numbers. In this way we can more easily find a routine in a long tape

should this be necessary.

AS. At the end of the tape after the last significant character there should be a tai I, consisting

of about 6 inches (15 cm.) of blank tape, followed by half'wdozen Erase characters (Er) and a very short

piece of blank tape. The little group of Er characters right at the end of the tape is very conspicuous

and effectively prevents our feeding the wrong end of the tape into a tape- reader.

A6. Blank tape should not appear except where indicated above. Isolated figure shifts (4^'s) are

allowed in a few places only; these are described below (in the rules for punching directives) .

The reader should particularly note rule A3 above; if CR LF is omitted after blank tape the computer will,

in most cases, stop when reading the tape. Incidentally, it will also stop if the wrong end of the tape

is placed in the tape-reader.

To avoid repetition in describing how order-pairs, numbers, and directives are punched we can state
two general rules applicable to all of them.

Bl. The character CR should always be followed immediately by LF; the combination is regarded as a

composite character and is denoted by CR LF.

B2, The character Er may be punched anywhere except between CR and LF.

The character CR is not one of the 16 odd-parity characters; in fact ER can turn into it by the loss of
a single hole, and CR can similarly turn into © + 6 or Sp. As a check, therefore, the Initial Orders
require that the character immediately after CR should be LP (an odd-parity character). The Er
character (Erase) has all five holes punched and can be used for correcting punching errors, especially
those which the operator realises he has made at the moment he presses the key. t

We shall now describe the rules for punching orders.
CI. Orders must always be punched as order'-pairs; single orders cannot be read in.

Each order of a pair is punched in the same way, as governed by the following rules. Note that circles,
boxes, vertical lines, etc. are never punched; they are merely guides in reading the programme sheet.
C2. The N-address in an order is punched as written, and followed by enough spaces to make the total

number of characters up to four (e.g. 2.0 Sp or 12 Sp Sp or 5 Sp Sp Sp or 2 + Sp Sp). No space is
necessary if N has four characters, provided the last is + (e.g. 0.7+ or 100+), if it is not + then a
single space should be punched (e.g. 1023 Sp).

The effect of this rule is generally to make the //-address have a width of four characters on the print-
out; the orders will then be printed in well laid-out columns for easy checking. The last character
of A^ must be either Sp or +, so that in the exceptional case when A^ is a decimal number of four or more
digits we must put in a single Sp after it (this is likely to occur only in pseudo order-pairs but may
also occur in single-word transfer orders in which the address is written in decimal - see Sec. 3. 10).

t He can back-space the tape, cover the incorrect characters with Er, and then punch the correct ones.
If he should press a key other than LP after punching CR then he must erase both the incorrect
character and the CR, and then punch CR LP.

129

6.5 INPUT AND OUTPUT

C3. The X-address, function digits and modifier of an order are punched as written; no spaces are

allowed between them. The modifier need not be punched (or written) if it is zero.
CU. Every order must be followed immediately by CR LF.
The following are a few illustrations of these rules:

Order as written Order as punched Order on print-out

4.0 6 01 4.0SP601CR LP 4.0 601

10 2 43 10SpSp243CR LF 10 243

l+[l]72 l+SpSpl72CR LP 1+ 172

5.0 6 00 4 5.0Sp6004CR LP 5.0 6004

1. 2+ 3 63 1. 2+363CR LP 1. 2+363

I 100 -I 70 100SP-70CR LP 100 -70

71 1000SP-71CR LP 1000 -71

1000

In fact it is permissible to punch spaces before the /V-address; but this is not recommended. The

following special rule applies to null orders, i.e. the order which is written as simply on the

programme sheet and which represents the dummy order 00.

C5. A null order must be punched CR LF; no spaces are allowed between the and the CR LF.

The fullowing rules govern order-pairs.

C6. Order-pairs are made up of the a-order followed immediately by the b-order, both orders being

terminated by CR LF. Before the a-order LF and CR LF may be inserted if desired.

C7. If an order-pair is a stop order-pair then a full-stop (@) should be punched after either the

a-order or the b-order, but not both . In this order the modifier must be punched, even if it is zero .

If just one of the orders of a stop order-pair is modified we normally punch the stop after that one.

Otherwise we usually punch it after the a-order. If a modifier is, zero it can, if desired, be punched

as Sp but this is not recommended.

Pseudo order-pairs (see Sec. 3. 12) are punched according to the same rules as order-pairs, but
they usually require more care in punching. Often the "6-order" in a pseudo order-pair will be an
unsigned integer (see Sec. 5.5), when the following rule applies.

C8. If one of the "orders'* in a pseudo order-pair is an integer it should be punched as written and

followed immediately by CR LF; no spaces are allowed after the integer.
Such an integer is stored by the Initial Orders at the least-significant end of the order.

The following are some examples of the way order-pairs may be punched.

Order-pair Order-pair

as punched on print-out

1.4SP724CR LP 1.4 724

15SpSp500CR LP 15 500

0.2SplOOO.CR LP 0.2 1000.

2+SpSp372CR LP 2+ 372

229SP-300.CR LP 229 -300.

48CR LP 48

3274SP--00.CR LP 3274 --00.

122CR LP 122

Between the blocks of a programme there should be about two inches (5 cm. ) of blank tape. The punching
should be as follows:

Last order of block (a 6-order) ending with CR LF,

Blanlc tape CR LP,

First order of next block CR LP, etc.

See rules A2 and A3 above.

If a block of the programme is not completely filled with orders and numbers then the length of
blank tape may be punched after the last order or number in the block. There is no need to fill up the
block with dummy orders or numbers, but care should be taken that the last order punched is a b-order
(i.e. the orders must be punched as order-pairs). Such a partially filled block would normally be followed
(after the blank tape) by a directive.

The rules for punching numbers are quite straightforward. Two kinds of numbers are recognized by
the Initial Orders, integers and fractions; they are distinguished from one another by the presence or
absence of a decimal point (punched as a full stop ©).

130

Ord

er-

pair

wrltten

1.4

0.2

00 0.

229

30 0.

3274

-0 0.

122

INPUT AND OUTPUT 6.5

Dl. Every number is introduced by its sign (+ or -), which must always be punched.

A number occupies a whole word in a programme, whereas an order is only half a word; the conversion

from the punched form into the internal binary form required by the computer is quite different for

orders and for numbers. The sign, punched before any digits, initiates the correct processes.

D2. Every number must be immediately followed by either CR LF or Sp.

Thus the terminating character for a number may be Sp or CR LF, whereas orders must be terminated by

CR LP (see Rule C4). The reason for allowing Sp as the termination of a number is that numbers are

often arranged in small groups; this grouping can conveniently be reflected in the layout of the

printing.

D3. Between the sign and the terminating character of a number the only permissible characters are

the decimal digits of the number, erases and a single stop (if the number is a fraction) ; no spaces are

allowed between the sign and the first digit or among the digits (if desired Er may be used to group

the digits).

D4. Between numbers (i.e. between the terminating character of one number and the sign of the next)

the following characters may appear: Sp, LF, CR LF, Er. Blank tape (4>) is also allowed provided it is

followed by CR LF (punched before the next number).

D5. The absolute value of an integer punched in the tape may not exceed 27^8779069^3 (i.e.. 2^^ -1) .

D6. The sign of a fraction is immediately followed by 0© and up to 11 decimal digits; the fraction

will he correctly rounded before being stored (exceptionally the fraction -1.0 is so punched).

The integer -2^^ should be punched as -1.0. Note that an integer may have up to 12 digits (left-hand

zeros need not be punched), but a fraction may not have more than 11 digits after the point. The

following are examples of numbers punched in accordance with these rules.

Numbers as punched Appearance on print-out

Blank tape CR LF

+92SpSp+0 CR LP +92 +0

Sp~6^Sp+4 CR LP -6 +4

-10SP-22CR LP LP -10 -22

+0.1234Sp~0.9876 CR LF +0.1234 -0.9876

+0.01122339 CR LF +0,001122339

-1.0 CR LF LF -1.0
Blank tape CR LP etc.

We shall now explain how directives are punched. A directive usually consists of a warning
character, which is a letter of the alphabet, and sometimes an address (see Sec. 4.3). We shall later
encounter directives which contain two addresses, and the rules for punching these are included below
for convenience of reference. Certain directives are in some way special, for example the N-directive,
which contains a name.
El. A warning character is always punched as follows: letter shift, warning character, figure shift.

For example a B-dlrective is punched \.B<t> and this is the complete directive, although it is usual
to punch CR LP before it.

E2. A single address in a directive should be punched as written, preceded by Sp (optionally) , and

followed by CR LF.

The space is helpful to separate the address from the warning character on the print-out but is strictly
not necessary. If desired extra spaces may be inserted to align the addresses in a number of directives,
but note that only a single is allowed between the warning character and the address. The siddress
must always be followed by CR LP. The following are examples of the punching of directives.

Directive as written Directive as punched

Z CR LF A. Z c;6

T 4.3 CR LF X.T<:;6sp4. 3 CR LP

E 2.2+ CR LP>vE0Sp.2.2+CR LP

T 2+ CR LPA.Tc;6sp2+ CR LP

E3. If there are two addresses in a directive punch the following after the warning character

Sp, first address, Sp, minus, Sp, second address, CR LF.

Here again the spaces are entirely optional but they do improve the appearance of the print-out. For
example the directive

P 14.6 - 15.3

Would be punched as follows,

CR LF X. P (?^ Sp 14.6 Sp - Sp 15.3 CR LF.

131 -

Directive

printed out

T 4.3

E 2.2+

T 2+

6.5 INPUT AND OUTPUT

E^. The following punchings are recommended for the names after the warning character Nj in both

examples the name is taken to be SYSTEM 3D. A length of blank tape must appear after the name,*

(a) Names of subroutines and other parts of a programme: blank tape CR LF h. N <p CR LF \. SYSTEM cp Sp 3

(b) Name of a complete programme: blank tape CR IF k N(p CR IF LF k SYSTEM (^Sp3kD4>CR LF 4>4>4> '"

In the above a sequence of 4>'s represents blank tape. Single figure shifts (</)) may be inserted where
necessary in the middle of a name. Two or more consecutive (ff s (normally punched as a short length of
blank tape) mark the end of the name; it is important to punch a length of blank tape after a name. If
this blank tape is omitted the next directive or piece of programme on the tape will be included in the
name and be simply printed out when read by the computer. A few extra LPs included in the name of a
complete programme serve to make it more conspicuous on the printed output of the computer and on the
print-out of the programme tape.

The reader is reminded that Er is ignored everywhere on the tape except between CR and LP (and in
names); so that an order may be punched, for example, as

Er 25 + Sp Er 2 Er Er 73 Er CR LP

and the directive T 5. 2 may be punched as

CR LP Er k Er T Er Er 5 Er. 2 Er CR LP

though these are rather extreme examples: The rule (A3) that blank tape should always be followed by
CR LP is relaxed with directives! strictly the rule should also allow k after blank tape without any
intervening CR LP, but it is advisable to follow the form of the rule given above. It is emphasized
that some of the above rules are recommendations which are not strictly binding - they can be regarded
as a guide to good practice, and will give an acceptable tape and a legible print-out. Many of the
rules must be followed precisely however, and departures are treated as punching errors; these will
cause a loop stop when the tape is read by the Initial Orders. This subject is discussed further in
Chapter 7.

A fairly common punching error is the omission of an entire order; this is sometimes not
detected when proof-reading the print-out against the programme sheet. The error will usually be
detected when the tape is fed into the computer, however, since it results in a block containing an odd
number of orders. The blank tape at the end of the block will therefore be read at a time when the
Initial Orders are expecting a 6-order and this causes a loop stop (it violates rules A6, CI, and C6 above),
The error will be detected at a slightly earlier stage if there are any numbers in the block after the
missing order; the Initial Orders will then encounter a sign (+ or -) when they are expecting a 6-order.

As an illustration of these punching rules we shall now describe the punching of the tape for the
programme "Special Pactorize" which was described in Sec. 4.4. The reader should compare this descrip-
tion with the programme sheet given in Pig. 4.3 and the print-out in Pig, 4.4. The tape starts as
follows.

Eight inches (20 cm.) of blank tape (leader) CR LP

kCKp Two inches (5 cm. ) of blank tape CR LP

A.N0 CR LP LP X SPECIAL cf^Sp k FACTORIZE CR LF

Two inches (5 cm. ) of blank tape CR LP

+ 420 CR LP

30 Sp Sp 640 CR LP

16 Sp Sp 610 CR LP

13 Sp Sp 640 CR LP

16 Sp Sp 610 CR LP

etc. . . .

The punching at the end of the first block and the beginning of the second is as follows.

7 ^ Sp Sp 401 CR LF (last order of first block)
Two inches (5 cm. ) of blank tape CR LP
1. 1 Sp 460 CR LP
16 Sp Sp 710 CR LP
etc. ...

Near the end of the tape the punching concludes as follows (starting here with the a-order in B5.6).

^ The name is taken as starting with the (f> after the N. The Initial Orders copy this and all

succeeding 4>'s until a non-blank character is found; the copying then continues to include the first
of two consecutive <?!>'s. For example, in the following sequence the characters underlined are printed
as the name.

CR LF X. N 0000c;6 CR LF LF X. SYSTEM <j6 Sp 3 >V D CR LF (/>0c/xj!)

Note that Er is not ignored in this process.

132

INPUT AND OUTPUT 6.6

Sp Sp Sp 0724. CR LP

0.0 Sp 060 CR LP

+100000000000 CR LP

Two inches (5 cm. ) of blank tape CR LP

^ E (?^ Sp 2. 1 CR LP

Six inches (15 cm) of blank tape and five Er' s
followed by one inch (2% era. ) of blank tape (tail).

When the tape has been punched it is spooled up.using a hand spooler (see Sec. 6.4). The name of
the tape, the date and the programmei'' s name are then written on the leader, and a rubber band is put
round the tape. If it is a short programme tape it is then put into a small white cardboard box; a
long tape would be put into a larger box. The length of the tape can be estimated in advance by allowing
about 16 inches (41 cm.) for each block, including a little blank tape.

6.6 The design ot subroutines for input and output

In most calculations input and output deserve study, particularly where there are large amounts of
either. It is always advisable to identify the input and output information in some way, preferably
automatically. The Initial Orders can provide some help here, if properly used. The date and serial
number and the name of each programme and subroutine can be printed automatically as the programme tape
is read in; these facilities should be fully used as the record they provide can be invaluable,
particularly during the development phase of a programme when errors are being detected and removed. '

Care should be taken to arrange the output material of a programme. so that it is as convenient as
possible to read. For example, it should be printed in columns of regular layout and divided into
blocks by extra line feeds at suitable intervals. Each number should be printed in the form in which it
is needed; all internal scaling factors should be i-emoved and the decimal point should be printed in
its proper place. It must be easily possible to identify each number; the layout can be helpful for
this, and consideration should be given to printing extra information, for example each line or block
of printing may be preceded by an identifying number (e.g. the value of some variable). It is often
helpful to print names at the heads of columns of figures, or carefully chosen symbols or letters in
front of some of the numbers. As a check on the calculation it may be advisable to print extra numbers,
such as check-sums. The use of only the odd-parity characters (see Sec. 6.1) and a regular layout of
the output matter can be very useful in detecting faults in the output punch. A single error in punching
or printing an odd-parity character will result in an obviously wrong character being printed or else in
a disturbance of the regular layout of the page.

A good way of checking the output tape is to use a special programme to read it in again and apply
certain checks, for example on the layout, the number of digits in each number and its approximate size,
the presence of special symbols, and the accuracy of check-sums. In some problems it may be possible to
repeat the whole calculation while the output tape is being read in and to check every character on it;
this kind of check is particularly useful when the output time is large compared with the computing time
since output can be suppressed during the checking run. A check like this is thorough but, of course,
the labour of doing it may not be warranted in most problems.

Both output and input subroutines should be carefully written or selected from Library routines.
They should be appropriate to the programme and as easy as possible to use correctly. An output sub-
routine should yield results which are easy to read. An input subroutine should have punching rules
which are as simple and as natural as possible. An output subroutine for general use should be
accompanied, if possible, by an input subroutine capable of reading in the output tape, complete with
all its layout characters. All special cases should be considered when the subroutine is being written.
For example, what are the effects of reading or printing the fraction -1.0? When printing fraxstions,
the addition of a rounding constant may cause overflow. It is vital that special numbers such as these
should not cause misleading results on input or output.

An input subroutine should give accurate results and incorporate as many checks as possible on
the punching. As each character is read in it will normally go through a chain of tests; these should
be arranged to reject impossible combinations of characters. For example a subroutine for reading in
signed fractions should indicate a failure if it reads more than one sign or decimal point, or if
overflow occurs. It is advisable to include as many checks as is reasonably possible on the input to a
programipe. The hand-punching may well include a number of errors. A particularly powerful check is
provided by check-sums, which can be the sum of all the numbers in any convenient group. The numbers
can be added, regardless of their significance, on a desk machine and the sum written on the data sheet,
for example at the foot of each column of figures. This sum can be punched with all the numbers, and
read in and checked by the programme. The programme should also count the numbers as they are read in,
if it is known how many there should be. It should also check in some way that it has been supplied with
the right tapes in the right order; this can be done by copying names on to the output tape or by using
certain identifying numbers or symbols punched in the input tapes.

The punching rules for the Initial Orders (see Sec, 6. 5) should be taken as standard. So far as
possible or appropriate an input subroutine should be so written that its punching rules are similar to
them or are adapted from them to suit particular circumstances.

Any subroutine should be adequately described in a specification. This would probably be sketched
out before the subroutine is written and all necessary details included afterwards. In an output
subroutine it would be necessary to include in the specification a fair amount of detail about the
exact printing produced in all cases. The specification of an input subroutine would include the
punching rules for the tapes designed to be read by the subroutine and a description of the effects
(e.g. loop stops) which occur if the rules are violated. The aim of the specification is to provide
full information about the use of the subroutine in such a form that the user does not have to refer to
the detailed programme.

T This operation is sometiraes known by the picturesque, if inelegant, name of debugging!

~ 133 -

▼

6.6 INPUT AND OUTPUT

We described in Sec. 6.3 the way In which an integer may be read in and converted to binary; the
orders dealing with a digit may be as follows.

C(5) = t < 9

multiply partial result by 10

5 7 01 add new digit from X5

Here the integer is being built up in XI; X5 is used to receive the character (t) from the input tape
and we suppose that tjhe orders given above are obeyed when t - 10 is in X5. This is because the
character will probably have been identified as a digit by subtracting 10 from its value (via 16) and
testing the sign of the result. After obeying the above orders we return to read the next character.

If the number being read in is not a decimal number (e.g. it might be a sum of money in sterling)
we can often use an adaption of the above sequence in which the partial sum is not necessarily multiplied
by 10 but by the appropriate radix.

In order to read in a decimal fraction we need a different action when a digit is read. The
general principle is to read the digits in as if they were the digits of an integer, but we simultaneously
build up a power of 10, which starts as 1(=100) when the decimal point is read and gets multiplied by 10
each time a digit is read. Thus, if we read the number . 123 from the tape we build up (a) the integer
123 and simultaneously (b) the integer 1000(=103). The value of the fraction can be found by a rounded
division when the end of the number is reached. For example, suppose initially that C(0. 0) = and
C(4) = 10 and that C(0. 1) is put equal to 1 when a decimal point is read. The following orders can deal
with a digit of the fraction, they are supposed to be entered with t-10 in X5.

(lO) 5 41 C(5) = t ^ 9

0.0 4 20 multiply pai-tlal result by 10

5 7 01 add new digit

0.0 7 10 replace result in 0.0

0.1 4 20
0. 1 7 10

• multiply C(0. 1) by 10

These orders build up an integer in 0.0 and a power of 10 in 0.1. Alternatively the multiplication by
10 can be done by a process of shifting and adding; this is sometimes desirable since multiplication
is a relatively slow process and it is preferable to take less than 3i^ milliseconds to handle each
digit (we can run the tape-reader at full speed). In the following sequence an integer (x say) is
built up in Xl and a power of 10 (y say) in X6. As before, we enter with t-10 in X5,

(Jo) 5 41 C(5) = * ^ 9

(T) 54 2y in X6 and 2x in XI

0.0 6 10 2y to 0.0

7 5 01 2a: + * to J5

(j) 54 8y in X6 and 8x in Xl

5 1 01 x' = lOx + t in XI

0,0 6 01 y' = lOy in X6

This sequence of orders is quite a good one; it resembles the one used in the Initial Orders. Note that
the sign-bit in XI is cleared by this sequence of orderst and does not affect the result (unless overflow
occurs). If therefore we set C(7) = -1.0 initially (instead of zero) we can test the sign to find out
if a digit has been read.

As an illustration of the filtering process to which each character can be sibjected as it is read
from the tape let us suppose that the following characters are possible, any other indicating a fault
(the values via register 16 are written below each character):

digits and + - © LP Sp Er CR
to 9 10 11 12 13 14 15 16 30

Having read a character we first subtract 10 from it to see if it is a digit, we then test for each
possibility in turn in the sequence given. Erase (Er = 15) is to be ignored but each other possibility
leads to some special action. A sequence of the following kind may be used.

* Because of a property of the double-length shift orders (see Sec. 3.7).

134

INPUT AND OUTPUT

6.6

Digit ^

+ 4k

-►16 5 00
@ 5 43

5 63

5 60

5 67

Action for

Read character value t

t-10

jump if a digit

jump if + (t=10)

form t-11 and jump if not minus

jt 5 67 form t-12 and jump if not ©

Action for ©

5 67

form t-13 and jump if not LP

Action for LP

d^ 5 67 form t-14 and jump if not Sp

Action for Sp

), ^ 5 67 form t-15 and jump if not Er
60 ignore Er

; ► 5 67 form t-16 and jump if not <j6
Action for 4^

^ 1^@ 5 43 form t-30

A—^C"^ 5 61 ^°P ^*°P ^^ ^^^ ^^
Action for CR

In this sequence we have not specified the action to be taken on each possible character, which
is bound to depend on the exact purpose of the subroutine. As a rule in each case there should be some
kind of a test that the character is permitted before the action is taken. After this there is in most
cases a jump back to the input order at the top to read the next character. This jump can be a
conditional one because of an interesting property of this kind of sequence; if the chain of 67-orders
is entered at some point with ^5 clear then ultimately the loop stop A will be encountered. The
conditional jump can therefore be an order which causes a jump only if the character is a permitted one
(for example, should not be allowed after a digit has been read).

A number of programming tricks can be used with input and output orders (see also Sec. 5.10). For
example, there Is the order

16 11

which has the effect of copying a character from the input tape on to the output tape. It reads the
number In 16 (i.e. the input tape character), adds C(0), and sends the result to register 16 (i.e. to
the output punch). This order is of rather limited usefulness since the character is not available
inside the computer for testing. In some programmes it may be desired to print a character or not
depending on certain circumstances; use can here be made of modified orders or conditional jumps.
Suppose, for example, that we can arrange that 5p = if no printing is wanted, or 5p = 1 if we have
to print CR LP. The following sequence can be used.

(30) 6 40

15 6 10 5

(13) 6 40

15 6 10 5

135

6.7

INPUT AND OUTPUT

Sending a number to register 15 (the handswitches) has no effect. Suppose again that C(5) is a small
integer which may be positive (or zero) or negative. Then the order

10 10 5

will print (h if C(5) is negative (when 5p = 7) and will do nothing otherwise. Under the same
circumstances the following orders will print the sign of C(5) (as + or -).

(22) 6 42 5

16 6 10

Here the first order puts -22 into A'6 if C(5) $■ and -21 into A'6 if C(5) < 0; these numbers are
congruent respectively to 10 and 11 (modulo 32) so that their five least-significant bits are those of
the sign characters. The following orders have the same effect,

(22) 6 40 5

16 6 12

If input via 17 is acceptable the following group of orders can be used to ig'nore a length of
blank tape and Erases,

— ^

0.0

17 6 02

0.0 6 60

_ , ,

(3?) 6 42 6
0.0 6 60

-t to 6^ (in 17 code)
jump if if- ( f=0)
~(31~t)^-^f--31 to 6c
jump if Er (t=31>

The code-conversion table-look-up technique described in Sec. 6.3 can be of great help in a
complicated input scheme where almost any character can occur. There can, for example, be several
tables which are used according to the stage which input of an item has reached, or according to the
▲ shift.

6.7 The monitors and control panels

There are three separate panels of controls on the computer. Located centrally in front of the
control desk (see Plate 2) is an inclined panel known as the programmers' switches or, simply, the
control panel. The keys or switches on this panel are the ones of chief concern to the operator and
the programmer; they are shown in Plate 11.. Above these is the vertical monitor panel on which are
visible the faces of the two monitor tubes and a number of associated controls (see Plate 12). The
third panel, known as the engineers' switches, is mounted under a hinged flap in the centre of the
control desk; it has been 'hidden away' because the controls on it are not often used and only a few
of them are of interest to the programmer or operator. The labels and numbers of the controls are
engraved in two colours, black and white; we are concerned only with those in white since the others
are intended for use by the maintenance engineers.

The most important controls are on the programmers' switches or control panel (Plate 11). The row
of 22 keys along the top of this panel are known as the handswitches, and were briefly described in
Sec. 2.9. These keys are numbered (from left to right) -2, -1, 0, 1, 2, ..., 19. Of these, the 20

keys numbered 0, 1, 2 19 can be sensed by the programme; they control the digits HO to //19 of

special register 15 (key is coloured red and is the sign-digit key of the handswitches). The two
keys numbered -2 and -1 are not available in register 15; they are used in some manual operations
(see below). The 19 keys 1, 2, ...,19 are divided into four groups labelled N, X, F, M to correspond
with the four parts of an order (see Sections 2.6 and 3.12); this has been done chiefly to facilitate
manual operations.

The .Start key and the Run key are located centrally under the handswitches. The Run key is
described in Sec. 3.9; it can be used to stop the computer, or to cause a single order to be obeyed,
or to allow the programme to continue after a 77-stop or an optional stop. The Start key is
described in Sec. 4.3; ' it selects the source of the orders which the computer obeys. In Its NORMAL
(central) position these orders are selected from the ordinary registers in the computing store in the
usual way. In the START (upper) position the start order-pair (see Sec. 4.3) flows into the order-
register to call in the Initial Orders. In the MANUAL (lowe|;) position the computer obeys orders set up
(in binary) on the hand-switches; this is called manual operation and will be described below. Manual
operation is used chiefly to test the computer and also to read in the Initial Orders when the computer
is being commissioned.

To the left of the Start key are ten small neon lights which indicate the principal kinds of stop,
the state of the overflow indicator, the paper tape and magnetic tape busy signals, and an indication
when one of the engineers' switches is not in the normal position. The lights are labelled as follows: -

136 -

INPUT AND OUTPUT 6.7

Optional stop

Stop order (77)

Main store or Buffer parity failure

Computing store parity failure

Paper tape

Overflow

Writing with overflow

Unassigned order

Engineers' switch set

Magne^tic tapet

The stops are summarized in Table 3.1. The parity failure lights are discussed below.

At the extreme left of the control panel is the key which controls the hooter. If this key is
down the hooter will be sounded when most kinds of stop occur; in fact it will sound if the Run key
is up on any stop except a loop stop, an input busy stop or an output busy stop (the last can occur
only if the punch breaks down).

Next to this is a key labelled PUNCH ON BLOCK TRANSFERS. This key can be used in the development
stage of a programme; if it is down the computer operates normally (except for a reduction of speed)
but any 72- or 73-order causes the main store block-number concerned to be punched in binary. This
useful feature is described in Sec. 7.6.

The next key labelled STOP ON OVERFLOW is described in Sec. 2. U. The key labelled INHIBIT
OPTIONAL STOP is described in Sec. 3.9.

In the centre of the control panel are two lights which indicate whether the next order to be
obeyed is an a-order or a 6-order. These lights are useful chiefly if the computer is stopped or if
manual or single-shot operations are being performed.

The keys labelled MONITOR 2 SELECTOR and DRUM TRIGGER are used with the monitor tubes and are
described below.

Before describing the monitors we must first discuss the question of the parity checks in
Pegasus. We have already described the parity check applied to input and output (see Sec. 6.1). On
input this is really a programmed check since all that is done by the computer is to convert the
property of having an odd (or even) number of holes into the property of being not greater than (or
greater than) 15; the programme has to do the rest. On output reliance is usually placed on visual
checking of the printed material; it is normal to use only odd-parity characters for the bulk of the
output. Both the main store and the computing store are subject to a parity check. A word is
normally regarded as containing 39 binary digits, but there are in fact 42. three of which are not
accessible to orders. Of these three bits two are gap digits (normally both zero) and one is a
parity digit. The parity digit is immediately to the left of the sign digit. When a word is placed
in the store its parity digit is adjusted so that the word has an odd number of 1 digits (and,
therefore, also an odd number of O's). When a word is extracted from the store its parity is
automatically checked to make sure that it still has an odd number of I's. Should this check fail the
computer stops and one of the stop lights on the control panel is lit; there are separate lights for
the main store (drum parity failure) and the computing store (computing store parity failure), and in
addition the hooter will sound if it is switched on.

The parity of all words read from the main store by a 70- or 72-order is checked before the words
are put into the computing store; and conversely, when words are written into the main store by a 71-
or 73-order their parity is checked before they are written. When a word enters the mill from an
accumulator or one of the ordinary registers its parity is checked and the parity digit removed; when
the result of the order is put back into the computing store a new parity digit is automatically computed
and inserted into the word. For example, when the order

2.7 3 01

is obeyed the words in 2. 7 and ^3 are checked and the result in X3 has a new parity digit manufactured.
Note that in the order

2,7 3 00

both C(2.7) and C(3) enter the mill, even though C(3) is not used, and a parity check is applied to
both of them. When a word enters the mill its parity digit is replaced by a copy of its sign digit;
this is then used to detect overflow.

The parity of an order-pair is also checked when it is extracted from an ordinary register and
copied into the order-register to be obeyed.

These checks are highly effective in ensuring the early detection of faulty components or
incorrect adjustments in either of the stores. Their use is based on the fact that the most common
faults in a storage device result in the alteration of a single binary digit, or else in the replace-
ment of a complete word by a string of C s or 1' s (either of which have even parity). Complete reliance
may be placed on the functioning of the stores.

t The arrangement of the neon lights is slightly different on Pegasus 1. There are two paper tape

busy lights, input busy and output busy (except on machines with magnetic tape- which have a

magnetic tape busy light and no output busy), and there is no light to indicate that one of the
engineers' switches is set.

137 -

6.7 INPUT AND OUTPUT

It should be noted, however, that accumulators 6 and 7 are not checked In this way. The reason is
that these accumulators are part of the complex circuits associated with multiplication, division,
double-length shifts and the Justify and normalize orders; it was deemed impracticable to apply a
parity check to them. A parity digit is, of course, supplied whenever the accumulators are written into
the main store with a block-write order.

Whenever access is not required to the main store, the equipment used for the parity check In it Is
instead used to check continuously the parity of the address track on the drum. This track is used
chiefly to identify the addresses of words round each track of the drum and so to initiate main store
transfers. It is thus possible for the computer to stop and light the drum parity failure light even
when the programme is not using the main store.

The left-hand monitor tube (Plate 12) can be used to display the content of any accumulator or
ordinary register (and also a few other things chiefly of Interest to the engineers). The two rotary
switches below the screen and to the left select the register to be displayed. The upper switch
(N BLOCK) selects one of the blocks of ordinary registers in the computing store. When the lower switch is
set to one of the numbers marked N UNITS the register with the corresponding position-number is
selected and its content displayed. As shown in Plate 12 the display gives the word in ordinary register
1. 6. To display the content of an accumulator the lower switch is simply turned to one of the numbers
marked X, the setting of the upper switch need not be changed. The word displayed on the screen is
arranged as an order-pair with the a-order above the 6-order and the digits conveniently grouped. Each
1 digit appears as a short vertical line and each digit as a dot. To the left of the a-order are the
sign-digit (or stop/ go digit), the parity-digit and the two gap digits. Thus the display shown in Plate
12 gives the following word in f/1.6.

gap digits N X F M

i I • I • I . . I 1 ... i I.I o-order

11.11 ■.11 I . . . ! 1 ... b-order

parity sign or
digit stop/go digit

This represents the go order-pair: -

3.2 4 61 5
(27) 3 43

The parity digit is a 1 to make the total number of I's odd. Reference should be made to Sec. 3. 12 for
a detailed description of the binary representation of order-pairs. When reading an order displayed on
the monitor screen it is usually best to read the function part first since this determines how the
yv-address is to be read. Note that the parity digit and the gap digits cannot be touched by the
programme.

It is possible for the display on the monitor screen to be in seven groups, each of six bits, for
use with mixed radix conversion operation. This display is given by the "Character" selection of the
three-position engineers' switch labelled *Scale' , which is just above the monitor tube.

The right-hand monitor tube can be used to display the content of the order- register, the
order-number register or the previous order number only (these can also be inspected on the left tube);
the three-way key labelled MONITOR 2 SELECTOR determines which of them is visible. The display is
generally similar to that on the other monitor. As shown in Plate 12 the order-number is 2.3, i.e. the
computer was about to obey the order-pair in 2.3 when it was stopped. The order-number or previous
order-number is displayed in the position corresponding to the iV-address in the a-order; its left-most
digit is always and is not used to indicate whether an a-order or a 6-order is to be obeyed next; this
is shown instead by the two lights in the centre of the control panel. The right-hand monitor tube is of
use only when the computer is stopped, otherwise the display is a blur. The content of the order-
register depends on whether the computer, when stopped, was about to obey an a-order or a b-order. The
display is as follows in these two states:

(a) When the computer is about to obey an o-order the whole order-pair is in the order-register.

The upper line of the display shows the modified o-order which is about to be obeyed, and the
lower line shows the unmodified b-order.

(b) When the computer is about to obey a 6-order the a-order is no longer in the order-register.

The upper line shows the modified fa-order which is about to be obeyed, and the lower line

shows the original unmodified b-order.
It must be emphasized that it is only in one of these two states that the computer can be stopped.
The upper line of the display always shows the next order in the form in which it is about to be obeyed,
i.e. modified. The Af-digits in the order are still visible and are held in the order-register, but they
have already been used to select the modifier and have no further effect. The stop/go and parity digits
have already served their purpose and are not present; they have been replaced, together with one of
the gap digits, by the 3-bit extension of the /V-address which results from the modification (see
Sections 5.3 and 5.8). Supposej, for example, that there is the following sequence of orders in the
computing store.

- 138

INPUT AND OUTPUT

6.7

3.5

3.6

2.7

„_

When the computer obeys the 6-order in 3.5 it stops. The next order to be obeyed is the a-order in
3.6 so that, if the display is examined when the 77-stop occurs, we shall find that the order-number is
3.6, the next order is an a-order, and the content of the order-register will be the modified a-order
from 3.6 on the upper line and the unmodified 6-order on the lower. If 4jx = 256.0 the order-register
display will therefore appear as follows.

extended /V- address

^ A ^

used modified
digits

gap digit

modified
a-order

1 unmodified
lb-order

If a single-shot is now given by moving the Run key down to SINGLE SHOT and releasing it. the a-order
will transfer J5266 into f/4. After this the 6-order will get modified and, in its modified form, will
replace the a-order. Thus, if 5/^ = 63.6 the following display will result.

extended ^■- address

. A .

gap digit

modified
6-order

unmodified
6-order

The lights will show that a 6-order is to be obeyed next. When obeyed the order is effectively

2.5 6 63

The sequence of events would have been exactly the same if the 77-order in 3.5+ had been omitted and
the order-pair in 3.6 had been a stop order-pair. If the 6-order in 3.5 were another order causing a
stop (e.g. an unassigned order, or a 73-order encountered with OVR set) then the machine would stop
in the state described above, with the order-number equal to 3.6, an a-order next, and the modified
a-order on the upper line of the order- register display.

Along the top of the monitor panel are three sets of neon lights. The seven lights on the left,
labelled EXTERNAL CONDITIONS, show which of the external-conditioning relays are switched on. These
are the relays controlled by the 74-order (see Sec. 3.10); in a basic Pegasus installation the right-
most light (no. 7) shows which of the two tape-readers is in use. The next six lights, labelled
SELECTED TRACK, show which part of the main store is in use. Each track on the drum contains 16 blocks.
If a block number be divided by 16 the quotient is the number of the track containing the block and
the remainder is the number of the block on the track. For example B243 is block number 3 on track 15,
since 243 = 15 x 16 +3. The lights always show the number of the track containing the block or word
used in the last main store transfer; this is called the selected track. In most programmes these
lights are continually flickering as various parts of the main store are referred to. It is sometimes
useful to note the condition of these lights when either a programne failure or parity failure occurs.
The six neon lights on the right show whether the power is on, whether the drum is turning at the right
speed, and so on. These lights should all be on. The computer should not be used if any of them are off.

We can arrange to stop the computer at a certain stage in a programme and then use the monitors to
examine the contents of various registers, and perhaps work through a small part of the programme one
order at a time. This process is called peeping and can sometimes give information about why a
programme is not working which it would be difficult to get in any other way. On the whole, however,
peeping is to be discouraged strongly. It is very slow to go through even a tiny part of a programme
and it is very easy to make a mistake when reading the monitor tubes. It is preferable to use the
computer itself to print out extra information in a nwre convenient form, which can then be studied
at leisure; one way of doing this is described in Chapter 7. If a programme fails to work when it is
first put on the computer then the first thing to do is to hand over the machine to the next user,
after noting down the state if a stop has occurred. A considered decision can then be taken about
detecting the error, and the necessary alterations made to the programme. There is a tendency among
beginners to assume that if they cannot get their programme to work quite quickly then the computer is
faulty, which is only seldom so in fact.

The engineers' switches, under a hinged flap in the control desk, are only occasionally of use to
the programmer. Here are some of the buttons used for switching on the machine, some keys and knobs
which should not be touched, and the following sets of keys which are sometimes useful:

(a) keys for inhibiting stops (77-stop, unassigned order, write-wlth-overflow, and parity failure),

(b) a key which prevents the main store being written into (inhibit drum write), which has one

or two uses.

139

6.7 INPUT AND OUTPUT

In various places around the computer are some red emergency buttons to shut off the power should an
accident occur.

We shall now describe the technique of manual operation; it is not often needed and should always
be used with great care since it is easy to go wrong. By manual operation we mean the technique of
getting the computer to obey orders set up on the handswitches; it should not be confused with manual
directives, which concern the Initial Orders; this is quite a different matter and is described in
Sec. 7.5.

When the Start key is down, in the position marked MANUAL, the computer takes its orders from the
handswitches, instead of from the ordinary registers as it does when the key is at NORMAL. In all manual
operation it is advisable to watch the order-register on the right-hand monitor tube and the next order
lights on the control panel. Orders should normally be obeyed one at a time, i.e. by pressing down the
Run key to SINGLE SHOT and releasing it (this operation is called giving a single shot). The order to
be obeyed next can always be determined by reading the upper line on the monitor tube; the lower line
will always show a dummy ordert while manual operations are being performed, and it may be disregarded.

Let us suppose the computer is stopped when it is about to obey an a-order, and that we then set up
an order on the handswitches (using digits -2, -1 and to extend the iV-address; there is no stop/go
digit in manual operation). If we now move the Start keytt to MANUAL then the order-pair in the order-
register will be replaced by a manual order-pair, in which the a-order is that on the handswitches and
the 6-order is a dummy. Giving a single shot now causes the a-order to be obeyed and the dummy 6-order
to be moved up; the next order lights will indicate a 6-order. When another single shot is given the
dummy order Is obeyed and a new order- pair enters the order- register; this will be a manual order-pair
if the Start key is still at MANUAL, but will be taken from the computing store if the key is at NORMAL.
If we leave the Start key at MANUAL we can obey a whole sequence of orders from the handswitches in
this way, giving two single shots to obey each one. We must set up the handswitches afresh whenever a
6-order is shown on the lights as next order; this setting will enter the order-register on the next
single shot and will be obeyed on the one after that. *

If we wish we can return to normal operation by returning the Start key to NORMAL after the last
manual order has been put into the order- register or when it has just been obeyed. The process of moving
the Start key to NORMAL does not of itself affect the order-register but it ensures that the next order-
pair to enter the order- register comes from the computing store. Provided there have been no manual
jumps the programme will be resumed at the point where it was originally left, since the order-number is
preserved during all manual operations (except jumps).

As an illustration of the technique we shall describe the procedure to transfer blocks 81 and 82
of the main store into blocks 4 and 5 of the computing store and them to jump to the order in 4.6+. We
must first put the Run key to STOP and we shall suppose that the computer is then about to obey an a-
order. The next step is to set up the order 81 072 on the handswitches, thus: -

We then put the Start key to MANUAL and this order (with a dummy 6-order) enters the order-register,
where we can examine it on the right-hand monitor tube. Next we give a single shot and the order is
obeyed, copying B81 into ^4. We now set up the order 82 El 72 on the handswitches. A single shot
puts it into the order-register and another one causes it to be obeyed, copying /?82 into t/5. We then
set up the order 4.6+ 60 on the handswitches, thus: -

When we give another single shot this enters the order-register and we then return the Start key to
NORMAL. We can now go to RUN, but it is probably advisable to give one or two single shots and
inspect the order- register to make sure that the proper orders are being obeyed.

If the computer is about to obey a 6-order at the time it is stopped the procedure is similar in
general, but one of the single shots at the beginning should be omitted. In this case it is
advisable to finish a sequence of manual orders with a jump, as was done above, to make sure that the
computer starts obeying normal orders at the right point.

As the reader will probably have realized, the whole procedure is somewhat complicated and there
are many opportunities to make mistakes. There is still one trap to be described; this concerns
modified manual orders. The need for modified manual orders is rare and they are best avoided, but for
those who wish to try them we must point out that a manual order which is put into the order-register
by the a<3t of putting the start key to MANUAL is not modified (i.e. its M-digxts are ineffective). In
other words the first of a sequence of manual orders is not modified; later orders in the sequence will
be put into the order-register when giving a single shot and these will be modified in the usual way.

A simplified procedure for manual operation in which a-orders and 6-orders are irrelevant is
probably best for general use. Here we simply stop the computer and go through the following three
steps for each manual order:

t Actually the order 0,0 00, not a null order. ,

tt The Start key should never be moved unless the Run key is at STOP.

i** The fact that two single shots must be given for each manual order allows us to set up and obey
orders such as

15 3 00

which can be put in with one single-shot (6-order next) and obeyed with another (a-order next), when
the handswitch setting can have been changed.

- 140

INPUT AND OUTPUT 6.7

(a) set up the order on the handswitches,

(b) put Start key to MANUAL and then back to NORMAL,

Here step (b) puts the order into the order-register and step (c) causes it to be obeyed. We repeat the
above steps as often as required; we can finish in one of two ways, either we call in the Initial
Orders by going to START, or the last manual order must be a jump. Note that if this simplified procedure
is used then none of the manual orders may be modified.

As a rule all manual operation is to be avoided, it can lead to waste of much computer time, it is
very easy to make a mistake, and when a mistake has been made it is often difficult to determine what
happened, as there is no record of the orders obeyed. It is in fact a stage worse than the extensive
use of the monitors, which cannot at any rate lead to a wrong result. These remarks apply to any high-
speed computer. It is much better to control operations by the input tape where this is at all possible.
To this end there should be a keyboard perforator near the computer and reserved for the exclusive use
of the operator. We shall describe in the next chapter some of the facilities which are provided in the
Initial Orders for changing a programme or stopping it in the middle or for printing out the contents
of parts of the main store. Manual operation is usually best left to the maintenance engineers, for
whom it is mainly intended.

There is perhaps one application of manual orders which is very useful, reasonably simple to
operate, and which yields results which might otherwise be very difficult to obtain. If a programme
stops or produces meaningless results it is often helpful to know the content of every register in the
computing store. Perhaps the best thing to do is to copy these contents into a convenient place in the
main store by obeying a sequence of 73-orders manually (using the simplified procedure); the Initial
Orders can then be used to print out the results (see Sec. 7.5).

Manual operation combined with the use of the monitors can allow us to inspect the content of any
storage location in the main store should this be necessary. In general it is best to use the Initial
Orders to print out such information but there may be exceptional occasions when this is impossible.
The simplest technique is to transfer manually the word or block required into the computing store and
use the monitors to examine it. If none of the computing store is available we can display the word
in the main store by switching the left-hand monitor tube to the position marked SELEXTTED TRACK. We
must then stop the computer and select the appropriate track by a manual 72-order and use the DRUM
TRIGGER keys on the control panel. The procedure is best illustrated by an example. Suppose we wish
to examine the word in B243.5. This word is in position number 5 of block number 3 on track number 15
(243 divided by 16 yields a quotient of 15 and a remainder of 3). We therefore stop the computer and
set up 3.5 (i.e. 0011 101) on the DRUM TRIGGER keys, and switch the left-hand monitor to SELECTED
TRACK. We now set up on the handswitches an order to read any block of track 15 into U6 (a non-existent
block), for example the order 240 |6]72 which is set up as follows on the handswitches.

track number always the same

/ ' ^r

This amounts to setting up the track number on the left-hand six keys of the handswitches (i.e. the keys
numbered -2, -1, 0, 1, 2, 3) and setting the other keys as shown. We then obey this order by putting the
Start key down to MANUAL and back to NORMAL and then giving a single shot. This operation selects track
15 by reading B240 harmlessly into (76. The left-hand monitor tube now shows the required word. This pro-
cedure may be used to monitor any location on the drum, but cannot be used to look at a word in the delay line
storage.

The first 16 blocks of the main store (or 32 blocks in some installations) are held on delay lines. In
order to examine the contents of this store, the following procedure must be used: -

1. Set the monitor selector switch to SELECTED TOACK.

2. Obey the manual order 6 72 to select track or 16 6 72 to select track 1. In these orders
tracks and 1 represent the main store blocks 0-15 and (optionally) 16-31 that are held on delay-lines.

3. Set up the required block address in the N-address position on the handswitches.

4. Go to MANUAL and back to NORMAL. (This selects the appropriate block and there is no need to single
shot).

5. Select the required word by setting up its position number on the last three drum trigger keys.

6. To strengthen the image on the monitor screen, the three left-hand drum trigger keys should be UP.
In connection with the use of the monitor tubes it is occasionally useful to know that some

unassigned orders clear certain registers before causing the computer to stop. The effects may be
summarized as follows.

07 x' =

17 n' =

30-36 n' =0 (these orders are unmodified)

47 x' =

75 no effect

- 141

Chapter 7

The Initial Orders

In this chapter we give a full account of the directives and their principal uses, and also a
description of some of the various sections of the Initial Orders, including Binary Input. We also
describe some ways of detecting programming errors, and the facilities which have been provided in the
computer and the Initial Orders to help detect and correct them.

7. 1 General description

The Initial Orders are a programme which permanently occupy 128 blocks starting at B896 in the
isolated part of the main store. t Their primary purpose is to read in programmes for the computer. They
can be regarded as made up of a number of more or less separate routines or sequences. One of these is
called Assembly, and it has the function of assembling a complete programme from a master- programme and
subroutines; it is described in Chapter 8. Another important section is called Input; this is respon-
sible for reading in order-pairs and numbers from the tape and placing them in the main store. tt Most
of the other sections of the Initial Orders are concerned with reading in directives (see Sec. 4.3),
decoding them, and taking the appropriate action.

The Input section is operating during nearly all the time that a programme tape is being read in;
it is left, in fact, only when a letter shift (A.) is read. The character X. always introduces a directive
on the tape. These directives are important features on a tape, since they control the operation of
the Initial Orders. So as to provide a record of the directives it is arranged that many of them cause
some informative printing as soon as they have been read from the tape; this is known as optional
printing (or optional punching) since it can be suppressed if it is not wanted. For example, the
T-directive, which is used to set the Transfer Address (see Sec. 4.3), causes optional printing of a
letter T followed by the new value of the Transfer Address; this printing is preceded by CR LP so that
it will appear at the beginning of a new line. By inspecting the printing we can find out the value of
the Transfer Address at any stage of the input of a programme, and hence determine where the various
sections of the programme were stored. Whether optional printing occurs or not is determined by the
setting of the redl sign-digit key of the handswitches. If this key is up (HO = 0) optional printing
occurs; this is usually done with programmes that are under development, since the information given
by the printing may be of great value in finding programming errors. If the sign key of the handswitches
is down (/^O = 1) the optional printing is suppressed; this is normally done only with fully developed
programmes.

The T-directive and the others described in Chapter 4 are included in Table 7.1, which also gives
the optional printing; each directive in the table has been given a typical address, where appropriate.
The optional printing produced by an E-directive occurs before the 77-stop. Note that the printing
caused by N and D is not optional; it always occurs, regardless of the setting of HO. The optional
printing produced by a B-directive includes the new value of the Transfer Address; thus if a tape
carrying the two directives

T 24.3

is read in, then there will be optional printing as follows.

T 24.3

B 25.0

These directives leave the Transfer Address at 25.0 and the relativizer at 25 (see Sec. 4.6). The
B-directlve on the tape must not include an address, or there will be a loop stop.

An address in a directive is always a main store address; it may refer to a word, as in a
T-directive, or to an order, as in an E-directive. The address of a word is usually written in block-
and~position form (with a relative block-number if desired), but it may be written in decimal form;

' In the Pegasus 1 with small drum, the Initial Orders occupy about 60 blocks in the isolated store,
starting from B512.

ft Input reads in the programme tape slightly faster than one block per second.

142

THE INITIAL ORDERS

7.1

the following are examples.

24.6
2+,5
3+
709

word 6 in B24
word 5 in B2+
word in B3+
word 709 (decimal address)

Directive

T 14.3
E 13.5+

J 13.5+

Z
Y

Effect

Set T.A. = 14.3

Enter programme at 6-order in B13.5, after 77-stop, by
copying B13 into UO, and next three blocks into Ul,2,3
and then jumping to 0.5+

As for E but no 77-stop

Print the following name

Print the date, increase the serial number by 1 and print
it

Increase T.A. (if necessary) to the start of a block;
set relativizer equal to new block-number

77-stop
Optional stop

Optional
Printing

T 14.3

E 13.5+
J 13.5+
none

none

B followed
by new T.A.

none

Table 7.1 Summary of some important directives.

The address of a single order is usually written in block-and-position form, sometimes with a relative
block-number; an a-order address (but not a 6-order address) may be written in decimal form. Any of
the above examples can be interpreted as a-order addresses; the following are 6-order addresses.

24.6+

2+.5+

6-order in B24.6
6-order in B2+. 5

The address in a directive is always punched exactly as it is written. An attempt to use a 6-order
address in any directive requiring the address of a whole word, such as a T-directive. will result m a
loop stop. Note that the optional printing always gives, in block-and-position form, the actual
(absolute) address used. If, for example, either of the directives

T 1+.3 or T 91

is read when the relativizer is 10, then the optional printing will be

T 11.3.

The Y and Z directives simply cause an optional stop or a 77-stop respectively; this allows the
input tape to be changed if desired. Operating the Run key at one of these stops calls in the Input
section of the Initial Orders to read in more tape, exactly as though the directive had not been there.
There is no optional printing with either of these directives. These stops are sometimes useful m the
development stages of a programme when the programme tape may be in separate pieces, each of which can
be terminated by a Y or a Z. If this technique is used it is advisable for each piece of tape to be
headed by a name-sequence (i.e. an N-directive followed by a name); this helps to prevent feeding in
the tapes in the wrong order.

The 77-stop when Z is read can be distinguished from that when an E is read by inspecting the
order-number on the right-hand monitor tube and the "next order" lights; these show 0.6(6) for a Z and

4.0(a) for an E. ^u o i <

The Initial Orders are called in to read a programme tape by the Start operation: the Run key is
set to STOP, the Start key is pushed up to START and allowed to spring back to NORMAL, and the Run key
is then raised to the RUN position. This causes the start sequence of the Initial Orders zo be entered,
which immediately tests the five right-hand keys of the handswitches (keys H15 to i719); if these are
all up the operation is called a Normal Start. In a Normal Start the Transfer Address is set equal to
2.0 and the relativizer to 2; there is then optional printing of three asterisks, which consequently
appear at the top of the computer output, above the date and serial number, unless optional printing is
suppressed. These asterisks show that the T.A. and relativizer were set to their starting valuesT.

t The Reference Address is also set to its starting value by a Normal Start; this subject is discussed
in Chapter 8.

- 143 -

7.2

THE INITIAL ORDERS

When all this has been done the Input section of the Initial Orders is called in to read the tape. If,
on the other hand, the five right-hand keys of the handswitches are all down when the start process is
carried- out, then Input is immediately called in without the T. A. and relativizer being touched; there
is then, of course, no optional printing of the asterisks. Other settings of these five keys are des-
cribed below (see Sec, 7,5).

The current value of the Transfer Address is kept in Ub.l^f (i. e, in the modifier part of the last
of the ordinary registers) and can be inspected on the monitors while the tape is running in. When each
word has been read in by Input and converted to binary, it is placed in Xl; the Transfer Address is
copied from 5.7 into one of the other accumulators and there is then a modified single-word write order
to put the new order-pair or number into the main store; after this the T.A. is increased by unity in
B.lj^. The relativizer is kept in U5.6 as an Integer (i.e. at the right-hand end of the register).
Normally these quantities are changing too fast to be read but it is sometimes useful to examine them
when the tape has stopped; their final values are still in these registers when the programme is
entered, e.g. by an E-directive.

The Transfer Address is kept in 5.7^. The sign-bit of 5.7 is always 0; the counter 5.1^ is not
disturbed by a T-directive or a B-directive (or their binary equivalents, see Sec. 7.7). This counter
may be useful if the Initial Orders are called in as a subroutine (see Sec. 7.2) but it is cleared
automatically if Assembly is used (see Ch. 8).

7.2 The use of the Initial Orders by a programnie

The main purpose of the Initial Orders is to read in programme tapes, on which there may be
numbers as well as order-pairs. It follows that the Initial Orders can be used to read in lists of
numbers at the same time as the programme tape; becauKse, for example, there may not be enough room for
them in the main store. The programme can be so written that when the next list of numbers is required,
it calls in an input subroutine to read in the list. Instead of supplying a special input subroutine to
do this it is possible for the programme to call in the Initial Orders an a subroutine. .If this is done
then- the Initial Orders will be used with two quite different objects; firstly to read in the
programme tape and enter the programme, and secondly to read in son«2 numbers under the control of the
programme.

When a subroutine has finished its work it normally obeys a link to return control to the master-
programme, A special directive has been introduced to signal the end of the numbers being resid by the
Initial Orders, and to cause a link to be obeyed; this directive is made up of the warning character L
(for link) - there is no address.

The part of the master-programme which calls in the Initial Orders as a subroutine to read in some
numbers should do the following:

(A) set in 5,7^ the main store address where the first number is to be placed, i.e. the Transfer
Address,

(b) set a link in XI for return to the master programme when all the numbers have been read in,

This last operation m\.st be done with a modified 72-order. It is convenient to use special register 37
to provide the modifier; this contains the fraction 7/8, which corresponds to the word (896.0, 0) if
interpreted as a modifier and a counter. ' For example, to read a list of numbers into consecutive
locations starting at B38.7, the master-programme could contain the following sequence of orders.

0.0

0.3 1

5, 7 1
0.4 1

10
00

37 4

10 §

72 4

0.0

38 -

70 0.

2+|0|

0.5

Het T.A. = 38,7

set link in A'l
set 4;^ = 896.0
B90G to UO
j ump to .

}^ - (38.7. 0)
link, obeyed when L is read.

We are here using the subroutine entry to the Initial Orders, and the following sequence of events
will. occur. First, the Initial Oi'ders preserve the ticcumulators by writing them into BO; the Input
section Is then entered to read the tape. As each number is read from the tape it is, as usual, placed
in the main store in the location determined by the Transfer Address in 5.7;»f, and the latter is then
increased by unity. After the last number on the tape an L-directive is punched, the effect of which is
to set the accumulators from BO, and to obey the link, i.e. the order-pair in XI (and in BO. l)tt.

t On the Ktnall drum Pegasus 1 the 4096 Initial Orders may be entered as a subroutine by transferring B522
to UO and jumping to 0.0. In this ca.se the modifier for the 72-order is conveniently provided by
special register 33; this contains the fraction ',i, or (512.0, 0) as a modifier and counter. It will be
seen, therefore, that the piece of programme in the next example may be made to work with the 4096-word
store by changing the order in 0. 1+ to 33 4 00.

ft The link is obeyed in f/0,3.

- 144

THE INITIAL ORDERS 7.3

There is no optional printing associated with an L-directive. The list of numbers to be read in in this
way must be punched according to the usual rules for the Initial Orders (see Sec. 6.5); they are termina-
ted by the L (punched as CR LP A. L 0).

It is not, of course, necessary that the tape should have only numbers on it; since it is being
read by the Initial Orders any of the usual directives may be used. For example, it is a good plan to
have a name-sequence on the tape before the first number. It is also possible to set the Transfer
Address by a T-directive on the tape, instead of setting it in the master-programme. We can also read
in order-pairs as well as numbers, though we may then have to set the relativizer (in 5.6 as an integer)
as well as the Transfer Address, t

The final value of the Transfer Address is in 5.7^ (and that of the relativizer in 5.6) when the
link is obeyed. This may be of some use to the programmer, for example it is easy to check how many
numbers have been read in. Apart from the accumulators, the whole of the computing store may get
altered when the Initial Orders are called in. It is inportant that none of the numbers should be put
into BO. 1 if the L-directive is to be used (this is where the link is kept).

The subroutine entry to the Initial Orders, which we have just been describing, is to be carefully
distinguished from the start entry at B896. 0, which is the entry used when we call in the Initial Orders
by using the Start key. When the subroutine entry is used the accumulators are preserved in BO and Input
is then immediately called in to read tape. When the start entry is used the accumulators are not stored
away, keys //15 to H19 of the handswitches are tested (see Sec. 7.1) and, if these are all up, the T. A.
and relativizer are set to their starting values, there is optional printing of three asterisks, and
only then is Input called in. It is the start entry which is used in the 'Special Pactorize' programme
described in Sec. 4.4. It is also possible to call in Input directly to read a tape without using the
subroutine entry; if this is done the accumulators are not copied into BO but the effects are otherwise
similar to the subroutine entry. This direct entry to Input can be used if, for example, a link for the
L-directive has already been put into BO. 1 or if there is no L-directive on the tape. To call in Input
in this way we must transfer B903 to U3 and jump to 3. 1; this is described below.

The main application of the direct entry to Input arises in interludes. An interlude may be
defined as a short piece of programme which is read into the store along with a main programme, but
which is obeyed as soon as it has been read in (by a J-directive on the tape). It can, for example,
perform a small calculation or some piece of organisational work connected with the input of the main
programme. After being obeyed, an interlude calls in Input directly and is then no longer required; we
usually arrange that it gets overwritten by the next routine on the tape. In order to facilitate this,
two effects of the directives E and J have been provided which have not so far been mentioned,, Either
of these directives, just before entering the programme, sets the accumulators from BO and then sets a
special link in Xl; when this link is obeyed Input is directly called in. This link is self-modified
(see Sec. 5. 10) and consequently must be in XI at the moment it is obeyed; it is the following order-pair.

100 |_3j72 1
3. 1 60

The modifier part of this word is 803.7 (N. B. 803 = 100 x 8 + 3) so that the a-order i-eads B903
(N. B. 903 = 803 + 100) into [73. An interlude can be usefully regarded as a subroutine of the Initial
Orders, which gets called in at the appropriate moment during input of a programme to do some calculation
and then obeys the link in Xl. The tape may be made up as follows.

B set relativizer

Interlude tape

(reset T. A,
interlude

^ Q^ ^ .^^^^ ..... so that

' ■ is overwritten.

J 0+ enter interlude.

The fact that an E-directive sets the accumulators (except for Xl) from BO on entry to a programme
may be useful; we simply place in BO the words we want to have in the accumulators when the programme is
entered. This can be done as a part of the input process of the programme. This procedure is, however,
not recommended because most programmes use BO as temporary storage (for the accumulators) while they are
running; the originar words in BO consequently get overwritten and it is then impossible to restart the
programme by a simple E-directive. To restart the programme we would have to restore the original
contents of BO first; this would be a serious handicap in the development stages of the programme. It
is always advisable to arrange that a programme may be restarted by an E- or J-directive, If this is
reasonably possible.

7.3 The detection and correction of blunders

We use the word blunder as meaning a mistalie in writing a programme. It is useful to have a special
word for this kind of mistake in order to distinguish it from mistakes in operating procedure, rounding
errors, truncation and other errors due to the use of approximate formulae, incorrect data, faults in the
computer and punching errors in the preparation of tapes. Blunders arise for a variety of reasons; a
wrong order may be written inadvertently, a short sequence of orders may not do exactly what was intended,
the problem may have been misunderstood, the same register or storage location may accidentally be used
simultaneously for two different purposes, some feature of the computer or a subroutine may have been
misunderstood, and so on. When a programme is first put on the computer it will probably contain a

T If the tape to be read in has any binary-punched information on It. (see Sec. 7.7) it is essential that
the word in 175.5 should not be negative.

145

7.3 THE INITIAL ORDERS

number of blunders and these will all have to be discovered and corrected: in a large programme this may
be quite difficult. However, it is usually worth trying a programme out once first - after all, it may
be free of blunders! The process of discovering blunders and correcting them is called developing the
programme.

The symptoms produced by blunders are as varied as the blunders themselves; some will result in an
unexpected or irregular layout of the results, or perhaps all the numbers printed may be zero, or there
may be no results at all, or the programme may start to read tape or punch blank tape indefinitely, or
it may stop or go into a loop of orders and stay there. These are all fairly obvious symptoms, and are
usually not too difficult to correct. The most difficult blunders to put right usually produce results
which differ only slightly from the correct ones. A carefully worked hand solution of one or two cases
is a great help in developing a programme. Any differences between the results of the programme and
those of the hand computation can be investigated in detail; not infrequently the hand computation is
the one shown to be in error, but it is encouraging to get results from the programme which are known to
be right.

When a new or partly developed programme is put on to the computer it may produce some results or
it may stop in some way, A careful note should be made, with the help of the monitors, of where the
programme stopped (if it did) and whether OVR is set; it may also be useful to print out the contents
of the computing store (see below, and Sec. 7.5). This should all be done quickly and then the output
tape or the printed results should be taken away for study. This makes the computer available for the
next user with as little wasted time as possible. No attempt should be made to diagnose possibly compli-
cated and misleading situations manually on the computer; in the event of the programme not working the
machine should be vacated at the earliest possible moment.. It is advisable, however, that all programme
sheets and other relevant papers should be readily available at the computer during the development of a
programme.

Quite often a preliminary run will disclose a number of small blunders, and perhaps show that not
all is well. The small blunders or slips can probably be put right quite easily, by altering the programme
tape in some way with the tape-editing equipment or by using X-directives as described below. When
preparing for a further run of the programme it may be useful to alter the -programme slightly so that it
stops at a certain point. When this point is reached in running the programme it may be desirable to
note, for example, the contents of certain accumulators or perhaps to examine or print out intermediate
results from the main store which can be compared later with a hand computation. To this end the following
directives have been provided in the Initial Orders:

S to make a specified order-pair into a stop order-pair,

G to make a specified order-pair into a go order'pair,

F to print fractions,

I to print integers,

tK to print contents of a location as 6-bit characters,

P to print parts of a programme (as orders),

X to correct orders in a programme.

All these directives contain addresses which, as always, refer to the main store.

Suppose, for example, that we wish to make the programme stop when the order-pair in 19.6 is
encountered. We can take the programme tape and insert (by splicing, see Sec. 6.4), Just before the
E-directive at the end, a short length of tape bearing the directive S 19.6. When read by the Initial
Orders this causes the order-pair in B19.6 to be made into a stop order-pair, regardless of what it was
before; there is also optional printing of S 19.6. When this has been done the Input section of the
Initial Orders is called in to read more tape; here it will read the E-directive to enter the
programme. If we wish we can put a number of S-directives on the tape. In a similar way the G-directive
makes the order-pair in the specified location into a go order-pair, with optional printing of G
followed by the address. Note that the address after either of these directives is the address of a word,
not that of a single order; there will be a loop stop (in U3.3) if a 6-order address is punched after
either of them (e.g. S 12.4+ will cause this). If desired, these S- and G-directives can be left on the
final form of the programme tape since the only printing they cause is optional, and it is usual to
suppress optional printing during the input of a fully developed progranme being used for production.
Since, however, S- and G-directives are generally only required temporarily, it may be better not to
splice them into the main programme tape. If the programme is a large one it will have a long tape and
it may be inconvenient to get at the E-directive punched at the inside end of the roll. An alternative
procedure is best described by an example. Suppose a programme tape ends with E 2.0 and that we wish
to make the order-pair in 25.3 a stop order pair. We can run in the programme tape as usual until
it stops on the E 2. at the end. We now take the end of the tape out of the main tape-reader and
replace it by a short tape bearing the following directives only:

S 25.3
E 2.

We then call in the Initial Orders by a Normal Start operation, instead of just operating the Run key.
The above two directives will then be read, the usual stop will be encountered on the E, and the
programme may then be entered by operating the Run key.

t The K-directive is not present in Pegasus 1.

146

THE INITIAL ORDERS 7.3

Intermediate results obtained by a programme can, if desired, be examined on the left-hand monitor
tube. We described in Sec. 6. 7 how to use this tube to display words in the main store; this procedure
is, however, rather inconvenient, involving as it does, the setting up of addresses in binary on keys
and the reading of binary information from the monitor tube. This peeping is generally discouraged as
it is not only inconvenient, but also time-wasting and does not provide any automatically printed record
of the operation or its results. It is preferable to punch a short tape carrying certain directives
and to read this in with the Initial Orders when information from the main store is wanted.

The F- and I-directives cause the Initial Orders to print words in the main store as fractions or
integers, respectively. For example the directive F 83. 1 causes the word in B83, 1 to be interpreted as
a fraction and printed out on a new line; the fraction is printed, unrounded, to 11 decimal places and
is preceded by its sign. In a similar way the directive I 83. 1 causes the word in B83. 1 to be interpreted
as an integer and printed on a new line; left-hand zeros are suppressed (i.e. replaced by spaces) and
the sign appears Immediately before the first digit to be printed. It is, of course, up to the program-
mer to decide how any particular word should be interpreted and to use the appropriate directive. When
the printing is finished the Input section of the Initial Orders is called in as usual to read more tape.
It is quite possible, and in fact not unusual, to have several F- or I~directives on the tape. If
desired, either of these directives may be followed by two addresses separated by a dash (i.e. a minus
sign), in which case all the words from the first specified location to the second will be printed out
(the second address must not be less than the first). For example, the directive

F 43.6 - 44.2

will cause the five words in the locations B43.6 to J544. 2 to be interpreted and printed as fractions,
each on a line to itself. The first number is preceded by CR LP and each number is followed by CR LF,
An extra line feed is printed between blocks for convenience in. reading the results. There is a
certain amount of optional printing as follows:

(a) the printing begins with P or I (on a line to itself), and

(b) each number is preceded by its main store address.

Note that the printing of the numbers always occurs, whereas that of the addresses (and the warning
character) is optional. If optional printing is suppressed the output tape is suitable for reinput by
the Initial Orders. t in all cases the numbers are printed in a single column, the right-hand end of
the integers being vertically aligned. The F- and I-directives can even be used to print the final
results of a programme, but this should be regarded as an expedient for temporary use. The technique
is to arrange that the programme calls in the Initial Orders when it has finished (or it can stop and
the Initial Orders can be called in manually); we must put a tape bearing the required directives
into the tape-reader (these could be punched after the E directive on the main programme tape).
The P-directive is generally similar to the P- and I-directives but it prints part of the
programme in the form of orders. The address or addresses in a P-directive are those of individual
orders. For example, the directive P 2.2+ will cause the 6-order in JB2. 2 to be printed, and the
directive P 12.6+ - 13.0 will cause printing of all the orders from the 6-order in Z?12.6 to the a-order
in B13.0, inclusive. Each order occupies a line to itself, the first order being preceded by CR LF and
every order being followed by CR LP, with an extra LP between blocks. The optional printing is similar
to that produced by P or I:

(a) the printing begins with P (on a line to itself), and

(b) each order is preceded by its main store address.

The individual orders are printed according to much the same rules as they are punched for input. The
/V-address, for example, is printed in the way it is usually written on programme sheets+t; this is
followed by the J-address and function and by the modifier-address (or Sp if this is zero). The output
tape is in fact suitable for reinput by the Initial Orders, provided

(a) optional printing is suppressed, and

(b) only order-pairs are punched.

Note that relative block-numbers will be punched as they are stored, i.e. in their absolute form; this
is generally desirable as we get information always about the actual orders in the programme as obeyed.
The a-order of a stop order-pair is followed by a full stop. As with F and I the Input section of the
Initial Orders is called in to read more tape as soon as the printing is finished. The P-directive is
very useful in telling us the details of the programme as obeyed by the computer.

As an illustration suppose that when the programme stops we wish to print the fraction in 3225
(decimal address), the integers in B29. 7 to 530,2, and the orders in B9.7 to BIO. 1+. We first prepare
the following steering tape:

F 3225

I 29.7 - 30.2

P 9. 7 - 10. 1+

t It will have to be provided with a length of blank tape at each end, which can be obtained by

pressing the run-out button on the output punch. Exceptionally the integer -238 cannot be read In
(see Sec. 6.5).

ft There are a few small and unimportant departures from the usual punching rules described in Sec. 6.5.
The address in a single-word transfer order is always printed in block-and-positlon form. A decimal
yV-address (e.g. that of a special register or in an order of groups 4, 5 or 7) Is aligned on the
right instead of on the left, i.e. such an address is preceded by enough spaces to make its width
equal to 3 and then followed by a single space.

147

7.3

THE INITIAL ORDERS

This tape is punched as follows (see Sec. 6.5):

Blank tape CR LP

-'\.P0Sp3225 CR LP

Blank tape CR LP

-Vl.'^Sp29.7Sp - Sp 30.2 CR LP

Blank tape CR LP

.\P.fSp Sp 9.7Sp - SplO. 1+ CR LP

Blank tape CR LP

'VZ<^ CR LP

Blank tape, erases.

When the programme stops we place this steering tape in the main tape-reader, clear the handswltches,
put HO (the red, sign-digit key) down if we wish to suppress optional printing, and then call in the
Initial Orders (by a Normal Start) to read in the tape. The computer then produces an output tape
which yields a print-out similar to those shown in Pig. 7. 1; here the left-hand column shows the
printing with optional printing; the right-hand column gives the result if optional printing is
suppressed. At the top is a print-out of the steering tape.

I 29«7 - 30*a
P 9.7 - io«i+

F Z

403»i +o«i3345678900

39»7 +123456769012

ZOmO

-987654

30,1

+993

30.2

+o,ia3456789oo
+123456789012

-987654

+992

9«7 10 173
9.7+1 •2+060

10 17a
i»2+o6o

io«o 3*o 4012*
io.o+4»3 500
io«i 4 740
io»i+ 5 720

3.0

4*2

4
5

Pig. 7. 1 Typical output from F- , I~ , and P- directives.

40Z2*

500

740

720

The K-directive causes the comiMiter to print the contents of the specified address (or addresses)
as 6-bit characters. It operates in the same way as the P- and I- directives. The content of a location
is taken to consist of six 6-bit characters laid out as follows: -

Number of bits

Contents

I
1

1 3

6 1

where

S indicates spare bits

Cj is the i-th character printed.

As far as possible the teleprinter characters printed by the K-directive correspond to the letters
or numbers which would be printed, using a standard 6-bit code, if the characters were sent to a line-
printer. Letter shift and figure shift characters may appear on the paper tape as well as the 6 printing
characters, but the word always terminates in figure shift.

The code is given in Table 7.2. Characters not listed are printed on the teleprinter as * (asterisk).
This code is the same as that used in the Perranti Converter (Magnetic Tape ♦— >• Printer or Cards) for
Pegasus and Mercury, except that on the converter the following four characters are different:

000000

010000

100000

110000

148

THE INITIAL ORDERS 7.3

If desired, the K-directive at a given installation could be set permanently to use a different
code.

Figure 7. 2 shows the print-out of a steering tape to print the contents of two locations with a
P-directlve and a K-directive, followed by the output: (a) with optional printing in the left-hand column,
(b) without optional printing in the right-hand column.

P io,3 - io»4+
Z

\tJ *^ *i/ \(/ M" \ly

4k 'i'* i^K 'K -i^ *i*

10.3 8 41 73«

10.3+ 10003 84173.

io»4 40002* IC)0 03

lo«4+72075 40002,

K 10,3 A3X103 A3Xt03

10.4 80587Q 80587Q

Pig. 7. 2 Typical output from a K-directive

If a blunder is discovered in a programme it is often possible to put it right by use of the tape-
editing equipment. The simplest kind of blunder may be called a slip; for example, writing the wrong
address in a jump order, specifying the wrong accumulator in an arithmetical order, or writing the
wrong function (a common slip is to write 00 instead of 40, e.g. to write(lO) 7 00 instead of(:lO) 7 40).
Usually a slip can be put right by simply changing a single order. A slightly more complicated blunder
may involve the rewriting of a part of a block because the orders actually used may not do quite what
was intended. This may be troublesome because room may have to be found for one or two extra orders;
considerable ingenuity and the use of conventional tricks (see Sec. 5. 10) may be needed. If we insert
a few dummy orders in each block when writing the programme, these can be a great help later. Often a
small sequence of orders may have to be moved up or down bodily so as to provide the space to insert the
extra orders; this will usually require changes in the addresses in jump orders. If room cannot be
found for any extra orders that are needed we may be able to patch the programme, i.e. replace an
order-pair by a block-read and a jump order to bring in and enter a spare block from the main store,
in which can be placed any orders needed. If a portion of a programme has to be rewritten it is
possible to punch new versions of the blocks containing the blunders and to splice them into the
programme tape in place of the original blocks. Short lengths of blank tape left between the blocks
can be a help here, and we may be able to reproduce part of the new blocks by reproducing from the
original tape. It is undesirable, however, to tamper extensively with a long programme tape; it may
not be easy to find the right section of tape, and one will have to read the tape-characters to
identify the blocks (actually not at all difficult after a little practice). It is thus easy to
Introduce extra errors, particularly as there may be a fair amount of unwinding and re-spooling to do.

Usually the best way to correct a small blunder is to use a small tape carrying a correction
sequence which can be fed in after the main programme tape and has the effect of correcting the
programme in the main store. In other words we read the original wrong programme into the computer
and then put it right. A number in a programme is simply altered by a small tape headed by a T-directive.
For example, a tape such as the following may be run in by a Normal Start operation after the main
programme tape has been read in (but not entered).

T 17.2

+ 443

E 2.0

The best way of correcting orders is to use the special X-directive which has been provided for
this purpose; with it we can easily change single orders or small groups of orders. An X-directive
containing the address of a single order is followed immediately on the tape by an order, which will
replace the one whose address is specified. For example, suppose we wish to change the 6-order in
B28.4 from 0.3 6 62 to 1.3 6 62; we can use the following directive:

X 28.4+

1.3 6 62

We can alternatively have two addresses after the X, when a number of orders read from the tape will
replace those in the main store between the specified locations (inclusive). For example the following

149

7.3

THE INITIAL ORDERS

directive will replace orders from the 6-order in B8.6 to the a-order in B9.0 by the orders punched
in the tape:

X 8. 6+ -

9.0

0.4 1

0.0

1.7+ 6

Character

Card Code

6 - Bit

Teleprinter

U.C.

L.C.

Character

Zero

000000

000001

000010

000011

000100

000101

000110

000111

001000

001001

Word Inhibit

Not

Aval

lable

001010

)

Space

010000

Erase

010001

010010

010011

010100

010101

010110

010111

011000

011001

Block Inhibit

Not

aval

lable

011010

(

Pull Stop

100000

100001

100010

100011

100100

100101

100110

100111

101000

101001

Paper Throw

Not

available

101010

Hyphen

110000

110001

110010

110011

110100

110101

110110

110111

111000

Minus

111001

Line Feed

Not

available

111010

Table 7.2 6-Bit Character Code for K-Directive

Each order is punched in the usual way (actually according to the rules for punching a-orders) so that
this directive would be punched as follows:

Blank tape CR LP

A.X0 Sp 8.6+Sp - Sp 9.0 CR LP

0. 4 Sp 120 CR LP

9 Sp Sp Sp 072 CR LP

0. Sp 060 CR LP

1. 7+663 CR LP
Blank tape

- 150 -

,6+ -

o«

130

073

o«

060

. 7+663

THE INITIAL ORDERS 7.4

When an X-directive has finished making corrections it returns to the Input section of the Initial
Orders to read in more tape; the Transfer Address will not have been altered. The optional printing
associated with an X-directive is very full and informative. First there is a letter X on a line by
itself; then there is a line of print for each order replaced, consisting of the address of the order,
the old order and the new order. The first line is preceded by CR IF, and each line is followed by CR LP,
with an extra LP between blocks. This is illustrated in Pig. 7.3; on the left is a print-out of the
X-directive given above, and on the right is an example of the computer output with optional printing.

9*° 8»6+o«4 131 o«4 120

8»7 9172 9072

8«7+090+o6o o»o 060

9»o i«7+662 i»7+663

Pig. 7. 3 Print~out of an X-directive and the optional
printing produced by it.

It optional printing is suppressed (HO = 1) there is no printing at all, and the relevant sections of
the Initial Orders have been made as fast as possible. This means that any corrections made by X-
sequences can be spliced into the programme tape and left there in the final form of the tape; they
will be read in at high speed. This is better than attempting to edit the programme tape when develop-
ment is finished because of the possibility of introducing errors; any. edited tape must always be
carefully tested again.

In an X-directive the orders are read in one by one and the corrections made; this is the only way
in which individual orders (instead of order-pairs) can be read in by the Initial Orders. An a-order
can, if desired, be marked with a full stop to make the order-pair a stop order-pair; a stop punched
after a 6-order will have no effect. It is advisable to punch a short length of blank tape after a
complete X-directive; this will help to prevent errors due to punching the wrong number of orders and
assists in splicing the tape. The optional printing is very valuable since it provides a complete
record of every order which was cha.nged. Furthermore, it shows the original form of the order as well
^ as its changed form, so that it is easy to undo a correction at a later stage and also to make sure that
any correction was put into the right place.

Relative addresses may be used in X-directives but great care should be taken that the relatlvizer
Is correct. For example, suppose we wish to alter the fe-order in B1+. 3 of a subroutine starting in B12
so that it reads 2+ 1 72. We can put the following directive at the end of the subroutine tape.

Xl-f. 3+

2+ 1 72

Alternatively we could put the following directives at the end of the programme tape.

T 12.0 I

> set relatlvizer =12.
B J

X 1+.3+

2+ 1 72

A sequence of this kind will, of course, change the Transfer Address.

Further help in detecting blunders is provided by the computer Itself by means of the facility
of punching on block-transfers. This is best used in conjunction with another directive, the
?-directive (read as "query-directive*'), which is not normally punched on tape. The process is
described in Sec. 7.6 below.

7.4 Summary of the directives 011 tape

Table 7. 3 summarizes the effects of the various directives when they are read from the tape by the
Initial Orders. In the column headed addresses indication is given of the kind and number of addresses
which normally follow the warning character. A single whole-word address is generally written as a,
and if two such addresses are permissible they are written aj - 02. The addresses of individual orders
are indicated as a(+) or as ai(+) - a2(+). For the sake of completeness a number of directives are
included which are described in later sections of the book. The details given in the table are of
necessity abbreviated, and the reader should refer to the sections indicated in the reference column for
a fuller description.

The directives E, J and L cause the Initial Orders to be left and a programme to be entered. All
the other directives (except ? and AG) cause the Input section of the Initial Orders to be called in to
read more tape when they have done their work.

The directives P, G, I, S and T cause a loop stop (after optional printing) if they are followed
(erroneously) by a 6-order address, e.g. T 17. 3+. A B-directive has similar effects if it is read
with a negative word in 5.7. These and other loop stops in the Initial Orders are Included in Table 7.7
in Sec. 7.8.

The <?S- directive has been included in Table 7.3; this directive is punched X.00 and can consequently
be punched in a length of blank tape simply as X. The effect is the same as Z, i.e. a 77-stop. This
directive can conveniently be inserted manually with a unipunch (see Sec. 6.4) in any stretch of blank
tape where a stop is desired.

- 151 -

7.4

THE INITIAL ORDERS

The character X. and all other letter shift characters not mentioned in the Table 7.3 are unassigned
directives, they cause a loop stop in 0.5 after optional printing (except that there is no optional
printing for A.).

If desired, an E- or J-directive may be followed by two addresses; the second address is disregarded
but will be found in 5.4^,^ when the programme is entered; this is occasionally useful.

Addresses

Effects

References

Optional
Printing

none

Enter Binary Input.

7.7

none

Prepare to use Assembly.

8.2

none

Change to second tape«reader.

8.2

none

Process the programme.

8.2

A3 and
details of
routines

a or

«1 ' «2

Punch out the word in a or the words
in a^ to ag in binary.

7.7

none

tA5

none

Prepare to read MAGLIB.

+A6

none

Search for required routine on MAGLIB.

none

Start new block with T. A. and set
relativizer.

4.6.7.1

B and new T.A.

Set relativizer to that of routine
number r.

8.8

C r

none

Print date and serial number after
increasing serial number by 1.

4.3

none

a(+)

Enter programme at a(+) in UO
after 77-stop.

4.3,4.6,7.2

E a(+)

a or

ai - ^2

Print fraction in a or fractions in
a^ to Oj.

7.3

P, addresses

Make word in a a go order- pair.

7.3

G a

a or
«1 " «2

Print integer in a or integers in ai

to 02.

7.3

I, addresses

a(+)

Jump to a(+) in UO.

4.3.4.6,7,2

J a(+)

a or Oj^ - a2

Print 6- bit characters in location a
or locations a^ to a^.

K. addresses

none

Transfer BO to accumulators, then obey
link in XI.

7.2

none

Print name of tape.

4.3,6,5

none

a(+) or

Print order in a(+) or orders in a.(+)
to a^(+).

7.3

P, addresses

ai(+) - a2(+)

Call for complete programme, number n,
on magnetic tape.

none

Reference. Add T. A. to next order-pair
and store in tag-list and not in T. A.

8.2

none

Make word in a a stop order-pair.

7.3

S a

Set T. A. equal to a.

4,3,4.6,7. 1

T a

a(+) or

Replace order in a(+) or orders in

7,3

X, addresses

ai(+) - a2(+)

ai(+) to a2(+) by orders read from tape.

old and new
orders

none

Optional stop.

7,1

none

77-stop.

4,4.7.1

none

Convert block-transfer tape.

7.3.7,6

<i>

none

77-stop.

7,4

none

Loop stop.

7.4

none

All 1
others]

Loop stop.

7.4

Letter

Table 7.3 Summary of the directives on tape.

t A5, A6 and Q are only available on machines which have magnetic tape; In the absence of magnetic
tape these three directives lead to a loop stop.

152

THE INITIAL ORDERS 7.5

f 7.5 Manual directives

When the Initial Orders are called in by use of the Start key the five right-hand keys of the
handswitches (//15 to H19) are tested, and their setting determines subsequent action. Usually these
keys are all up, a setting we shall describe as 0, and we enter Input to read tape after setting the
Transfer Address and the relativizer (and the Reference Address) to their starting values; this is
the Normal Start, which has optional printing of three asterisks. If the five keys are all down, a
setting we describe as Er, we enter Input immediately, and there is no optional printing.

If the five right-hand keys of the handswitches are neither all up (4>) nor all down (Er) when the
start operation is performed, then they are treated as if they were a warning character read from the
tape (via register 17 - the direct input). If an address is required then it is taken from keys HI to
/fl4; in this address /fl to HIO give the block-number, Hll to //13 give the position-number, and //14 is
used as an a/b digit in single-order addresses (A/14 = 1 indicates a 6-order address). HO is used
exclusively to control the optional printing. The keys H(-l) and H(-2) have no effect. The setting of
the handswitches thus determines a directive with a single address; it may be called a manual directive.
The way the keys are used may be shown diagrammatically as follows.

optional
printing

manual directive

no
effect

blocks

position (+) warning

character.

A ^,

-2 ~l'

4 5 6 7

8 9

10 11 12 13 14 15 16

17 18 19

In this diagram the numbers which are underlined correspond to red keys. As an illustration suppose
that the setting of the handswitches is as follows when the Start key is operated (keys marked 1 are
down).

-2-10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

000 000 001 001 101100 10 1

= 5 (+) =5 (E)

manual directive E 9. 5+

The five right-hand keys represent the integer 5 in binary; this corresponds to E, the fifth letter
of the alphabet. The address is 9.5+. The effect of the operation is therefore exactly the same as
if the directive E 9.5+ had been read from the input tape, and we get this printed, before the 77-stop
associated with an E-directive, because optional printing is not suppressed (^0 = 0). In this way we
can enter a programme which is already in the store. Note that the address which is set up is always
an absolute one, relative addresses cannot be used in manual directives.

With few exceptions (principally P, I, K, P, Q and X) any manual directives have almost exactly
the same effects as the corresponding tape directives. The main difference lies in what happens when
the appropriate action has been completed; most tape directives then call in the Input section of the
Initial Orders to read tape, whereas most manual directives cause a 77-stop, after which an operation
known as restart occurs and a new manual directive is taken from the handswitches.

For example, suppose there is a programme in the store and we wish to make the order-pairs in
B3.6 and BIO. 2 into stop order-pairs, and then enter the programme at B2.0. The procedure is as follows.

(a) Run key to STOP.

(b) Set up S 3.6 on handswitches (S is the 19th letter so the five right-hand keys are set to

10011; of the other keys //9 to H12 are down).

(d) Run key to RUN (the manual directive is carried out and there is a 77-stop).

(e) Change address in manual directive to 10. 2 (no change in five right-hand keys; of the other

keys HI, 9 and 12 are down).

(f) Run key to STOP, then to RUN (new manual directive S 10.2, then a 77-stop).

(g) Set up E 2.0 on handswitches (key H9 down and five right-hand keys set to 00101).
(h) Run key to STOP, then to RUN (77-stop on E-directive).

(i) Clear handswitches or set them as required by the programme and operate Run key to enter the
programme.
Note that the Start key need be operated only once (step (c) above), but the sequence of events would
be exactly the same if it were operated again after steps (e) and (g). This is because the restart
operation which occurs after the 77-stops in steps (d) and (f) is almost identical with the start
operation at the beginning of the Initial Orders.

We can now define more precisely the effects of the start operation, which is usually the
result of operating the Start key but may occur if a programme uses the start entry (B896 to UO, jump
to 0.0). The handswitches are immediately read. If the manual directive is (^6 (five right-hand keys
all up) then the Transfer Address, relativizer and Reference Address are set to their starting values,
there is optional printing of three asterisks, and Input is called in to read tape. If the manual
directive is Er (five keys all down) then Input is immediately called in with no other effect. Any
other setting of the five right-hand keys causes the appropriate manual directive to be carried out.

After most manual directives there is a 77-stop; when we operate the Run key to allow the
computer to continue the restart operation occurs. The handswitches are immediately read and
subsequent effects are the same as those in the start operation, except that the manual directive (p has
exactly the same effects as Er, i.e. Input is called in without the Transfer Address, etc. being

- 153 -

7.5 THE INITIAL ORDERS

altered. This difference has been introduced to facilitate the use of manual T- and B-dlrectives. For

example, to set the Transfer Address to 6.0 and then read in a tape the procedure is as follows.

(a) Run key to STOP.

(b) Set up the manual directive T 6.0 (T is the 20th letter so the five right-hand keys are

10100; of the other keys H8 and 9 are down).

(d) Run key to RUN (77-stop after T.A. is set to 6.0).

(e) Clear handswitches (except possibly for HO).

(f) Run key to STOP, then to RUN.

The last operation causes the tape to be read without any change being made in the Transfer Address.
Here we must not operate the Start key after step (e) or we shall do a Normal Start and the T.A. will
be reset to 2.0. A manual B-directive is similarly used; here the address set on keys HI to //14 is
entirely disregarded, as in any manual directive which does not need an address (e.g. D or L).

As was mentioned above, most manual directives lead to a 77-stop and the restart operation when
they have been carried out. Tlie principle exceptions (apart from and Er) are, of course, E, J and L,
which cause a programme to be entered. The manual R-directive is entirely without effect; it simply
leads directly to the 77-stop and restart: the R-directive read from tape is used with the Assembly
Routine and Is described in Chapter 8. The manual N-dlrective can be used to print a name from the
input tape; when the end of the name is reached there is a 77-stop and restart. This and the manual
directives Y and Z are of little value.

The manual directives P, I, P and X are very useful; the procedure for using them is rather different
from that for most manual directives. Suppose, for example, that we set up the manual directive P 16.4
and do a start operation. The fraction in 516.4 will be printed out as usual and there is then an optional
stop. If we operate the Run key at this stage, without changing the setting of the handswitches, then the
fraction in the next location B16.5 will be printed, and again there is an optional stop. This procedure
may be repeated Indefinitely. If optional stops are inhibited the Initial Orders will continue to print
fractions from consecutive locations in the main store until stopped, for example by raising the key for
inhibiting optional stops. The process continues in fact until the setting of the handswitches is changed
at one of the optional stops (this can only be done if optional stops are not inhibited); when this happens
the restart operation immediately occurs and a new manual directive is taken from the handswitches. The
manual I- and K-directives can be used in exactly the same way for printing integers or six-bit characters
respectively.

The following procedure can be used for printing numbers in randomly placed locations in the main store.
;a) Run key to STOP.

[b) Optional stop inhibit key UP (i.e. stops).

[c) Set up P (or I) and first address on handswitches.

[d) Start key to START and release it.

[e) Run key to RUN (first number is printed, then optional stop).
;f) Change address on handswitches to that of the next number.

(g) Run key to STOP, then up to RUN (next number is printed, then optional stop),
(h) Change address on handswitches to that of the next number.
:i) Run key to STOP, then up to RUN (next number is printed, then optional stop).

The last two steps are repeated as many times as necessary. When the last number has been printed we
can set up a new manual directive and operate the Run key. Note that at any time when the address is
being changed we can also change P to I or vice versa. There is the usual optional printing for P
or I provided HO = 0; this gives the address of each number printed and should not as a rule be
suppressed since it provides a valuable check on the correct setting of the addresses.

The procedure for printing out a series of numbers from consecutive locations in the main store
is as follows.

(a) Run key to STOP.

(b) Optional stop inhibit key DOWN (i.e. no optional stops).

(d) Start key to START and release it.

(e) Run key up to RUN.

Printing will continue. To stop it the best procedure is to raise the key for inhibiting optional
stops and then set a new manual directive, after which the Run key can be operated or the start opera-
tion carried out. As usual with P or I the numbers are printed in a single column with the blocks
separated by an extra line feed; if optional printing is not suppressed each number is preceded by its
address. The extra optional printing takes a certain amount of time and if many consecutively stored
numbers are to be printed it is best to suppress it. A useful procedure to make sure that the original
hands witch setting was correct is the following.

(a) Run key to STOP.

(b) Optional Stop inhibit key UP (i.e. stops).

(d) Start key to START and release it.

(e) Run key up to RUN (first number and Its address are printed, then optional stop).

(f) Suppress optional printing (HO = 1).

(g) Inhibit optional stops (key DOWN),
(h) Run key to STOP then up to RUN.

The last stop causes the first number to be printed again, this time without its address, and then the
printing of the following numbers starts. This is because the handswitch setting was changed (HO was
depressed) at the first optional stop at step (f). Consequently the restart operation occurs at step
(h) and the first number gets printed again. The optional printing with the first number (at step (e))
gives its address as set up on the handswitches; it is then easy to determine the addresses of subsequent
numbers, since they are grouped into blocks. The address of the number actually being printed can at
any moment be found by examining on the monitor tube the word in i/5,3; the modifier part of this word
gives the address. When it has reached or got beyond the last number wanted we raise the key for
inhibiting optional stops (alternatively we can put the Run key to STOP but this may stop the printing
in the middle of one of the numbers).

The manual K-directive operates in the same way as P and I. The manual P-directive is similar to
P and I except that individual orders are printed. If the handswitches are not altered at an optional
stop then printing of the next order occurs. If optional stops are inhibited consecutive orders are
printed out starting at the address set on the handswitches. The various procedures described above
for P and I can be applied also to P.

- 154 -

THE INITIAL ORDERS 7.5

The manual X-directive can be used to change orders in the store; it is in some ways similar
to P. I, K and P but there is in addition a 77-stop in the process, at which point a new order is set
up on the handswitches. For example, if we wish to change the 6-order in B15.6 to read 1.2 60 we
set up the manual directive X 15.6+ and do a start operation. There is than a 77-stop and we now set
up the order 1.2 60 on the handswitches, t and operate the Run key; this causes the order In the
main store to be changed and an optional stop occurs. If we operate the Run key at this stage, without
changing the setting of the handswitches, then the computer will prepare to change the next order in
the store, i.e. the a-order in B15.7, and there is a 77-stop to allow us to set up this next order. The
procedure may be repeated indefinitely. If we inhibit optional stops the computer will continue to
replace consecutive orders in the main store by orders taken from the handswitches; there is a 77-stop
each time to allow us to set up the next order. This process continues in fact until we change the
setting of the handswitches at one of the optional stops (this can be done, of course, only if the
optional stops are allowed to happen), at which point the rpstart operation immediately occurs and the
new setting of the handswitches is taken as a manual directive.

For example, suppose we wish to change the 6-order in BIO. 2 to read 12 6 40 and then enter the
programme by E 2.0. The procedure is as follows.

(a) Run key to STOP.

(b) Optional stop inhibit key UP (i.e. stops).

of the other keys //7, 9, 12 and 14 are down).

(d) Start key to START and release it.

(e) Run key to RUN (there is optional printing of a letter X; then a 77-stop).

(f) Set up the order @ 6 40 on the handswitches; the setting is as follows:

000 0001100 110 100000 000

(g) Run key to STOP and then up to RUN (the order is changed, there is optional printing of the

address, the old order and the new order, after which there is an optional stop),
(h) Set up the directive E 2.0 on the handswitches.

(i) Run key to STOP and then to RUN (there is optional printing of E 2.0 and then a 77-stop before
the programme is entered).
If desired the Start key could be operated just before step (h) above and the effects would be the same.
The following procedure can be used if we wish to change several orders in arbitrary places in the
main store.

(a) Run key to STOP.

(b) Optional stop inhibit key UP (i.e. stops).

(d) Start key to START and release it.

(e) Run key to RUN (there is a 77-stop).

(f) Set up first order on handswitches.

(g) Run key to STOP and then to RUN (there is an optional stop),
(h) Set up X and address of next order on the handswitches.

(1) Operate Run key (there is a 77-stop).

(j) Set up next order on handswitches.

(k) Operate Run key (there is an optional stop).
Steps (h) to (k) are then repeated as often as required. After each optional stop there is optional
printing of the letter X, and after each 77-stop the address, the old order and the new order are
printed optionally (i.e. if HO = 0) . When the last order has been changed (i.e. at an optional stop)
we can set up a new manual directive and operate the Run key. Note that at any optional stop we can
operate the Start key without affecting the results; this is indeed probably to be preferred as less
likely to cause confusion. It is unwise to suppress optional printing when carrying out manual X-
directives; the printing shows exactly what was actually done and it is then easy to put any errors
right.

The procedure for changing a sequence of consecutive orders in the main store is as follows.

(a) Run key to STOP.

(b) Optional stop inhibit key DOWN (i.e. no optional stops).

(d) Start key to START and release it.

(e) Run key to RUN (there is a 77-stop).

(f) Set up first order on handswitches.

(g) Operate Run key (there is a 77-stop).
(h) Set up next order on handswitches.
(i) Operate Run key (there is a 77-stop).

Steps (h) and (i) are repeated as often as required. At any 77-stop the address of the order about to
be changed can be read from the monitors by displaying C(5.3); the modifier part of this word is the
address and the sign-bit is a 1 if it refers to a 6-order. When the last order has been set up on the
handswitches we can raise the key for Inhibiting optional stops before operating the Run key; when the
next optional stop occurs we can set up a new manual directive. Alternatively we can break the sequence
by doing a start operation.

It should be noted that any X-directive always treats the stop/go digit of an order-pair as
belonging to the a-order; we should therefore, normally set HO = 1 when setting up an a-order; this
key is disregarded in 6-orders so it can safely be left down (after step (f), when it will have no effect
on optional printing) when changing a sequence of orders in the way described above. Another small
point to note is that it is necessary to change the setting of the handswitches in some way at the
optional stop before a new manual directive can be read. Exceptionally it may happen that the

t This order is set up on the handswitches as though it were to be obeyed manually (see Sec. 6.7); the
digits being set up on the groups N X F M ot the keys. The setting for this order is as follows (see

Sec. 3.12):

000 1001010 000 110000 000

N X F M

The Af- address in any such order must be an absolute one.

155

7.6 THE INITIAL ORDERS

handswitch setting for the last order is exactly the same as the setting for the new manual directive.
This is likely to occur only if a null order is put in last and it is then desired to read tape. It is
easy to avoid this difficulty by doing a start operation instead of simply operating the Run key
(alternatively we can use a manual cj6-directive with an arbitrary non-zero address).

On the whole, manual directives of any kind should be used sparingly, and then always with optional
printing. If the operator is in doubt about the procedure then only the simplest operations should be
used. The safest directives are P, I, K and P since they cannot affect the contents of the main store.

If a programme goes quite wrong and, perhaps, stops after a few seconds it is often possible to get
valuable information by printing out the contents of the computing store. The procedure Is to obey
manually (see Sec. 6,7) a sequence of orders such as either of the following,

(a) 073 or (b) [o]73

1 [T]73 2 [2]73

2 [2J73 3 (T|73

3 [3]73 1 \T\12

4 [4]73 5 [5]73

5 [5]73 4 [T)73

6 [ll^S 6 [T|73

These orders copy away the whole of the computing store, which can then be printed out, probably by the
manual directive P '0.0. (Optional stops should be inhibited, and the printing can be stopped at the
appropriate moment by consulting the modifier in U5.3. This contains the address of the word being
printed, as has already been described. ) Before doing this a note should be made of the state of the
overflow-indicator and of the order that the machine was about to obey. The overflow-indicator can be
cleared, just before obeying the above manual orders, by putting the Start key up to START before putting
it down to MANUAL. Note that if the manual orders are obeyed in the sequence (b) given on the right
above, fewer key changes are needed. It is usually best to use a P-directive for printing out the
computing store in this way because it is usually orders that one is interested in and the information
printed is complete, in the sense that the value of every binary digit can be determined. (It is easy to
find the values of modifiers and counters. )

In table 7.4 are summarized the effects of the manual directives. Some directives are included
which are described in later sections of the book. The optional printing is not given because it is
the same as that caused by the corresponding tape directives, except for the manual 0-directive which
has no analogue on tape (the tape (/>-directive is equivalent to Z). In this table the following
abbreviations are used:

RESTART The restart operation occurs.

77 RESTART A 77-stop occurs, followed by the restart operation.

TAPE The Input section of the Initial Orders is entered to read tape.

The setting of the five right-hand keys of the handswitches is shown by a five-digit group in which a
A represents a key which is raised and a 1 a key which is down.

▼ 7.6 Block -transfer punching

There is a key on the left of the main control panel (Plate 11) which is labelled PUNCH ON BLOCK
TRANSFERS. When this key is depressed the computer operates normally except that every time a block-
transfer order (i.e. a 72- or 73-order) is obeyed the computer will punch, in the output tape, the
number of the main store block concerned in the transfer. The punching consists of three characters for
each block-transfer order; the first character is always Er and the next two give the block-number in
binary (these two characters each give 5 bits of the 10-blt block-number - in the direct code used in
register 17). This punching is interspersed with the ordinary output (if any) of the programme being
obeyed; it results in a fair reduction of speed as a rule.

This facility is a powerful aid in the diagnosis of blunders; it is provided by special circuits
in the computer and is not connected in any way with the Initial Orders. Since, however, the punched
information is in a binary code, it cannot be easily read or printed out. A special facility has
therefore been put into the Initial Orders to convert the binary information into decimal; this is
done with the ?-directive as described below.

If a programme is known to be correct up to a certain stage, after which its behaviour is difficult
to account for, the facility of punching on block-transfers may provide a useful clue since the block-
transfer orders determine the contents of the computing store and control most of the vital transfer of
information to and from the main store. The following procedure, or some adaptation of it, may be used.

(a) Read the complete programme into the store in the usual way and then alter it (e.g. by an S-directive)
so that the computer will stop when the last correct stage has been reached.

(b) Start the programme as usual (e.g. by an E~directlve).

(c) When the computer stops because of the alteration made in the programme, tear off the output tape
and press the RUN OUT button on the punch to obtain a leader of blank tape.

(d) Press down the PUNCH ON BLOCK TRANSFERS key.

(e) Allow the computer to carry on with the programme (e.g. by operating the Run key). The computer
will now punch whenever a block-transfer order is obeyed.

- 156

THE INITIAL ORDERS

7.6

Last 5 keys

Setting

Effects

Reference

R
S
T
X

2
3
4

5
6

7
9
10
11
12
14
16

18
19
20
24

25
26
29
31

00000

00001

00010
00011
00100

00101
00110

00111
01001
01010
01011
01100
OHIO
10000

10001

10010
10011
10100
11000

11001
11010
11101

11111

All others

Normal Start: Set T. A. = 2.0, relativizer = 2, R. A. = 882.7, then

TAPE after optional printing of three asterisks.
Restart: TAPE (no optional printing)

Decimal digit is taken from the red keys if 11 to 13; keys

H4: to HIO should be clear (other keys are disregarded).

Effects as follows:

AO No use (enters Binary Input).

Al Prepare to use Assembly, then TAPE (after B operation).

A2 Change to second tape -reader; 77 RESTART.

A3 Process the programme, then TAPE (after B operation).

A4 No use (reads addresses from tape and enters Binary Punch).

A5 Search for Library on MAGLIB; 77 RESTART T

A6 Search for routine on MAGLIB;

Set T.A. to new block, set relativizer; 77 RESTART.

Set relativizer to start of routine specified; 77 RESTART.

Print date and serial number after increasing serial number
by 1; 77 RESTART.

Enter programme at a(+) after 77-stop.

Print fraction in a. After optional stop repeat with next
number from store if handswitches unchanged; otherwise RESTART.

Make order-pair in a a go order-pair; 77 IffiSTART.

Print integer in a. Otherwise as P.

Enter programme at o(+); no stop.

Print 6-bit characters in o. Otherwise as P.

Set accumulators from BO; obey link in XI (and 50.1); no stop.

Print name from input tape; 77 RESTART.

Print order at a(+). After optional stop i-epeat with next order
from store if handswitches unchanged; otherwise RESTART.

Read complete programme (number tapped out after optional
stop) from magnetic tape.

No effect; 77 RESTART.

Make order-pair in a a stop order pair; 77 RESTART.

Set T.A. equal to a; 77 RESTART.

Replace order at a(+) by new order read from handswitches after
77-stop. After optional stop repeat with next order in store
if handswitches are not again changed; otherwise RESTART.

Optional stop; 77 RESTART.

77-stop; 77 RESTART.

Convert block- transfer tape.

Leave T.A. relativizer and R. A. untouched; TAPE (no
optional printing)

Loop stop.

7.5

8.2
8.2
8.2

8.7
8.7

7.5

8.8

4.3

7.5
7.5

7.3

7.5
7.5
7.5
7.5
7.5
7.5

8.7

7.5
7.5
7.5
7.5

7.5

7.5
7.6
7.5

Table 7.4 Summary of the manual directives.

(f) Stop the computer manually when it is thought that enough information has been punched out (this
can alternatively be programmed).

(g) Press the RUN OUT button on the output punch to obtain a tail of blank tape, mark the end of this
tape and tear it off. This tape is referred to as a block-transfer tape.

(h) Return the PUNCH ON BLOCK TRANSFERS key to its normal position. It is important not to forget
this operation.

(i) Place the leader of the block-transfer tape in the main tape-reader. Press the RUN OUT button
on the punch for a moment.

(j) Set up a manual ?-directive by setting the five right-hand keys of the handswitches to 29 (i.e.
11101 in binary).

(k) Do a start operation to call in the Initial Orders. These will now read the block-transfer tape
and produce a new tape with the block-numbers in decimal as described below. The process stops when the
block-transfer tape runs out (or it can be stopped manually).

- 157

7.7 THE INITIAL ORDERS

This is the way in which a block-transfer tape can be obtained and converted to a readable form
with the aid of the ?-directive (read as "query directive"). This directive is almost always used
manually; it causes the block-transfer tape to be read and ignored as far as the first Er character;
the next two characters are then interpreted as a block-nuraber and printed in decimal (preceded by
CR LP) after which the next Er is sought. We therefore get a single column of the block-numbers; this
is preceded by optional printing of "?". If there is any punching on the block- transfer tape between a
binary block-number and the next Er then an asterisk is printed (preceded by CR LP); each asterisk
printed among the block-numbers therefore corresponds to one or more characters on the block- transfer
tape which will have been produced by the programme under test.

Note that if the normal output of the programme being tested contains any Er characters then
misleading results will be obtained when the block- transfer tape is converted. This is unlikely to
happen, however, since few programmes require to punch Er.

7. 7 Binary input and output

The ordinary way of writing and punching a programme has been designed largely for the convenience
of the user of the computer. It is clearly valuable to be able to prepare programme tapes easily and
to print them out and edit and splice them in various ways. These features are less important, however,
when a programme gets beyond the development stage and is used regularly for production work. The
programme is then treated simply as a length of tape which causes the computer to do a certain job and,
provided an adequate specification has been written, there is seldom any need to alter it nor to
enquire what orders it contains. t At this stage what matters is that the progratrane tape should be
complete and convenient to use, and that it should be short and capable of being read into the computer
in the least time possible.

These requirements can usually be met by converting the programme tape from its ordinary Initial
Orders form into a new abbreviated form; in fact we use the computer to prepare a new binary tape
carrying the whole of the original programme in a highly compact form. This new tape can be read into
the computer and is, for most purposes, the equivalent of the original tape; it is handled as though it
were the original tape except that it cannot usefully be printed out on the tape-editing equipment.
The words on a binary tape are punched in such a way that they closely resemble their form inside the
computer: roughly, each binary digit corresponds to a single hole on the tape so that a single tape-
character corresponds to 5 bits and eight characters are enough for a whole word (compared with the 18
characters needed to punch an order-pair in the usual way if both orders are unmodified). Such tapes
are unintelligible to a human being but they can be read in at high speed (nearly 5 blocks per second)
by a very simple input routine called Binary Input, which forms part of the Initial Orders.

There are two main programmes for preparing a binary tape, since it is, of course, almost impossible
to punch such a tape by hand. The Binary Punch routine is a section of the Initial Orders; it can be
used to punch out in binary a programme held in the main store; we shall describe below how It can
be used. The special programme known as Binary Translation reads a tape punched in the ordinary Initial
Orders notation and simultaneously punches out the corresponding binary tape. The tapes produced by
either of these programmes are handled as if they were in the Initial Orders notation since a special
directive (AO) appears at the start of the binary-punched information; this directive calls in Binary
Input. A complete programme is normally converted into binary by using the Binary Punch routine since
the result is a complete, self-contained tape. The Binary Translation programme is most often used to
convert Library subroutines into binary.

Binary Input has its own set of binary directives corresponding to the most important ordinary
directives (and a few others); they are of course punched in binary and are produced automatically as
required by the Binary Punch and Binary Translation programmes. The ordinary user of the computer need
not concern himself with them. The Transfer Address and relativizer (and the Reference Address) are
used by Binary Input in the usual way.

The A-directives must now be described; they are concerned with the Assembly routine, Binary Input
and Binary Punch, which are all sections of the Initial Orders. The letter A which introduces one of
these directives is always followed by a single decimal digit, which may be 0, 1, 2, 3, 4, 5 or 6. This
digit is considered as combined with the A to form a composite warning character; it is not an address.
The effect of an A-directive is determined by the digit. The AO-directive calls in Binary Input; it Is
always immediately followed by information punched in the special binary code required by Binary Input.
This directive is never punched manually, but is automatically put in where required when the binary
tape is prepared. The directives Al, A2, A3, A5 and A6 are normally used with Assembly and are
described in Chapter 8. The A4-directive calls in Binary Punch to punch out in binary some Information
held in the main store. This directive includes two whole-word addresses (as with P or I for example),
which specify the first and last words to be punched out. Por example, the directive

A4 2.0 - 23.7

causes the contents of B2 to B23 inclusive to be punched in binary. This directive would normally be
punched as follows tt.

Blank tape CR LP

X.Ac^4 Sp2.0 Sp - Sp 23.7 CR LP

Blank tape.

t It goes without saying that properly annotated programme sheets should be prepared for any complete
programme; these supplement the specification of the programme, which should give adequate and full
details of (a) the operating procedure, and (b) how the data are prepared, and (c) the form and nature
of the results.

tt Here all the spaces are optional but they contribute to the legibility of the print-out and should be
included (especially the Sp before the first address). The CR LP before the A is not strictly required
but that after the two addresses must be put in. It should be noted that the digit 4 must immediately
follow the A.Acp; no spaces are allowed between the A and the digit. As usual, Er characters may
appear anywhere except between CR and LP.

- 158 -

THE INITIAL ORDERS 7.7

The AO- and A4-directives must be read from tape and not from the handswitches; there is no
optional punching with either of them. If an A4-directive includes only one address then the binary
tape will contain only one word; this is sometimes useful. When the binary punching produced by an
A4-directlve is finished Input is called in to read more tape.

The binary tape resulting from the use of an A4-directive is suitable for reinput; when it is
read in each word will be put back into the location it originally occupied; there is no arrangement
for putting the words anywhere else. In fact at the start of the binary-punched section is an
AO-directive to call in Binary Input; immediately after this is the binary equivalent of a T-directive,
which sets the T.A. so that the information all goes into its original locations (this is usually what
is wanted in a complete programme). At the end of the binary information there is a checksum, which is
simply the sum of all the words and binary directives on the tape (disregarding overflow). This
checksum is punched automatically when the binary tape is prepared; while the tape is being read in by
Binary Input a new checksum is built up and this is compared with the one punched at the end of the
tape. If there is disagreement then a loop stop occurs (in 113. 4i), but otherwise Input is called in to
read more tape (in Initial Orders notation). A checksum disagreement is usually caused by a torn or
dirty tape but may be due to a malfunctioning of the tape-reader; it is usually necessary to read the
whole programme in again. This sensitive check is automatically carried out on all binary tapes.
T While the binary tape produced by an A4-directive can be used as it stands for reinput, it is

usually preferable that it should carry a few ordinary directives in addition to the binary information.
Presumably the binary tape is to replace the original programme t^e; it should therefore start with
a leader and a D-directive and an N-directlve at least; and it should finish with an E-directive and
a tall of blank tape and some Er characters. It is quite possible to splice all these on to the
"bare" binary tape, but it is preferable that they should be put in automatically at the time the binary
tape is prepared. This is quite easy to do by using N-directives for putting information on to the
output tape. Let us suppose, for example, that the programme to be punched out occupies locations
B2.0 to B23. 7 and also locations B35.0 to i?43.2 inclusive; its name is to be SYSTEM 4 BINARY and it is
to be entered by a directive E 2.0. We first of all prepare a steering tape as follows (horizontal lines
indicate lengths of blank tape terminated by CR LF).

becomes leader

■^ itja,

uei ui tjUKci jLiig uapc

and D-directive

becomes N-

directive

SYSTEM 4

BINARY

programme

'a4

2.0

- 23. 7
- 43.2 J

causes programme to be

35.0

punched in binary

becomes E-

'n_

directive

E :

!.0

becomes tail

fN_

and Er' s

Er Er

Er Er Er

. causes a stop

Er Er

Er Er Er ■-«

tail of steering tape

Most of the items on this steering tape correspond to something on the binary tape. The steering
tape is read by the Initial Orders and produces the binary tape; before doing this we must of course
put in the programme which is to be converted to binary. It is as well also to suppress optional
printing (HO = 1) and to tear off the tape at the output punch. Consider the sequence of events as
the steering tape is read. First there is an N-directive and this causes the following "name" to be
copied on to the output tape; the "name" here consists of a length of blank tape and a letter D.
Thus we get the leader of blank tape and the D-directive at the start of our binary tape. Next on the
steering tape is another N-directive; this causes copying on to the binary tape of the following
"name" consisting of N and the actual name. Then come two A4-directives to punch out the actual
programme from the main store. This is followed by two more N-directives, which copy E 2. and a tail
on to the binary tape. This steering tape could be punched as follows (the characters underlined will
be copied on to the binary tape - see the footnote to Rule E4 in Sec. 6.5).

Blank tape CR LP

X.N (/) blanks CR LF

kDcf) blanks CR LF

?vN c/) blanks CR LF

A. N0 CR LF LF

\ SYSTEM Sp 4 Sp ^BINARY CR LF

- 159

7.8 THE INITIAL ORDERS

blanks CR LP

X A<^ 4 Sp Sp 2. Sp - Sp 23. 7 CR LP

blanks CR LP

kA4> 4 Sp 35.0 Sp - Sp 43.2 CR LP

blanks CR LP

A.Nc6 blanks CR LP

^E0 Sp 2.0 CR LP
blanks CR LP
A.N0 blanks CR LP
Er Er Er Er Er- Er
blanks CR LP
kZcf) blanks CR LP
Er Er Er Er Er Er

The A4-directive causes an inch of blank tape to be punched before the AO and the binary information.
The above steering tape will produce the following binary tape.

Blank tape CR LP

\.D0 blanks CR LP

X.Nc;6 CR LP LP

A. SYSTEM cf) Sp 4 Sp A. BINARY (p CR LP

blanks CR LP

iVA^Oand binary information from jB2.0 to B23.7

blanks CR LP

A.A0Oand binary information from 535.0 to B43. 2

blanks CR LP

\E0 Sp 2.0 CR LP

blanks CR IF

Er Er Er Er Er Er

If the steering tape is run with optional printing (HO = 0); the only difference is that the binary
tape will start with three asterisks, which would have to be cut off. The sole purpose of the Z-
directive at the end of the steering tape is to cause the computer to stop when the binary tape has
been prepared.

The time taken to prepare a binary tape on the teletype punch is about 1 second per block of the
programme. A fairly short programme can thus easily be converted into binary. With a long programme
it may be advisable to produce a binary tape before development of the programme is complete; one
must try to balance the time saved by using a binary tape for input during the rest of the development
against the time wasted by preparing a binary tape. A binary tape is very convenient to use, and one
may be able to give up the possibility of printing out and editing one's programme tape at a later
stage of development. We can of course always splice corrections on to the end of a binary tape, and
these can quite well be left there in the final version of the programme if they are not too numerous
and extensive.

7.8 Detailed description of the Initial Orders

The Initial Orders may be usefully considered as made up of the following sections.

(a) The Start sequence, which is called in by a start operation; it will reset the T. A. etc. if
the manual directive is (p and give the optional printing of three asterisks. The restart operation
is also done here.

(b) A Decode sequence for reading warning characters and dealing with most of the optional printing.
This sequence is entered when X. is read by Input or when a manual directive other than is to be
carried out. It reads in a warning character and the following it and calls in Address Input if
required. It also carries out the simpler directives (B, G, L, N, R, S, T, Y, Z, (^) and most of the
X-directive and includes a few checks.

(c) Input. This reads the whole of the tape except for directives and binary-punched information.
It is called in by a Normal Start operation and, usually, when a tape directive has been carried out.
Input is left only when \ is read from the tape.

(d) Address Input. This reads in the one or two addresses in a tape directive. (The address is a manual
directive is read in by the Decode section. )

(e) Number Print can be used for printing fractions or integers. It is used by P, I and D (for the
serial number). It is also used to print the block-number in addresses in the optional printing for
most directives, and by Order Print to print decimal ^^-addresses.

(f) Order Print is used by P and also for the optional printing for X.

- 160

THE INITIAL ORDERS 7.8

(g) A'directives. This routine reads the single digit associated with A, does the optional printing
and calls in the appropriate routine. It deals completely with A2.

(h) C-directive. This calls in Address Input to read the routine number, does the optional printing
and sets the new relativizer (see Sec. 8.8).

(i) D'directive. This prints the data from B895. 7, increases the serial number in B895.6 by unity and then
calls in Number Print,

(j) E' and J-directives. This routine organises the transfer of four blocks of the programme into the
computing store, sets the accumulators from BO, sets a link in XI and enters the programme in [/O (the
77-stop associated with E is in the Decode sequence).

(k) X-directive. This is really only a part of the sequence of orders obeyed when an X-directive is
read; it replaces the order in the store. The optional printing is mostly done by Order Print.

(1) ?-directive. This routine converts the block-transfer tape, using Number Print to print the block
numbers.

(m) Next Operand. This routine is used by P, I, K, P and X to advance the process to deal with the
next number or order in the store and to test for the end. It deals with both tape and manual directives.

(n) Binary Input reads binary tapes; it is called in by an AO-directive.

(o) Binary Punch prepares binary tapes when called in by an A4-directive. It calls in Address Input

to read the addresses.

(p) Q' directive. This routine is used to read and place on the drum complete programmes which are

stored on the same magnetic tape as the library.

(q) Assembly. This is a complex programme which is described in Chapter 8; it may itself be regarded as
made up of a number of sections;

Al, A5, A6 and L entries,

Process tag- list,

Process cue-list,

Search index,

Exit,

Optional interludes,

Part 1 i

Part 2

Process programme (A3 entry).
Assembly printing.

There are a number of routines used in connection with magnetic tape and cards: these will be
discussed later.

It is desirable to describe the Input section in a certain amount of detail; it is then possible
to determine the effects of certain slightly unusual punchings (e.g. in pseudo order- pairs), should this
be necessary. While reading tape. Input may be in any of six different states, as determined by the
nature of the item being read in and the stage reached in reading the item. These states may be
numbered from 1 to 6.

Input is always entered in state 1, which is also known by the name 8-search (beta-search). In
this state the following characters are ignored:

Er, Sp, LP, CR LP.

Certain other characters determine the nature of the item to be read in; these are as follows.

Decimal digit (0-9): an order-pair.
Sign (+ or -): a number.
Figure shift (4> or blank): blank tape.
Letter shift (k): a directive.

Any other character is treated as a fault and causes a loop stop. If a decimal digit is read state 3
is entered, which deals with the decimal digits of orders. A sign leads to state 5, which deals with
numbers. A figure shift calls in state 6 (called the a-search), and a letter shift causes Input to be
left for the Decode section dealing with directives.

In state 2, Input searches for the first character of a 6-order, which should be a decimal digit
(0-9). The characters

Er, Sp. CR LP

are ignored and all other characters cause a loop stop. If a digit is found then state 3 is entered.

In state 3 the decimal part of the AT-address is read in. This part of the address is made up of
tjecimal digits; as each is read in, the partial address is multiplied by 10 and the new digit added.
This state is left when one of the following characters is read:

Sp (as in the AT- address of an order of groups 4, 5 or 7): the iV- address is now complete.

+ (as in a relative block number); the relativizer is added to the A^-address, which is then
complete.

161

7.8 THE INITIAL ORDERS

When any of the above characters has been dealt with, state 4 is entered to read the remaining (octal)
digits of the order. If CR LP is read it is treated as terminating the order (e.g. a null order
punched as (31 LP, or the counter in a pseudo order-pair consisting of a modifier and counter). In
this state only Er is ignored; other characters not mentioned cause a loop stop.

In $tat€ 4 the octal digits of an order are dealt with. When this state is first entered (from
state 3) an octal digit counter c is set equal to 4. As each digit is read the value of c Is reduced
by 1, the partially formed order is shifted up 3 places (i.e. multiplied by 8) and the new digit is
added. There should be either 3 or 4 octal digits after the A^-address. Apart from digits the following
other characters may be expected.

+ This normally occurs only in jump addresses (e.g. 3.7+); only the 6 least-significant bits of

the address are retained and c is reset to 4. The N- address is now complete.
Er This Is ignored,
Sp If c = this is ignored, so we can punch spaces at the end of an order.

If c = 1 this is taken as replacing a zero jW- address; the partial order is shifted up 3

places and c is made zero.
If c is neither nor 1 the Sp is taken as coming between the N- and X- addresses and c is reset

to 4.
If a minus sign is read the only effect is to reduce c by 1; the partial order is not shifted
up. (Thus all information before the minus appears 3 binary places to the right of its
expected position. )
© This indicates a stop order- pair provided it is read when c = 0.

CR LF This terminates the order; if c = 1 the partial order is first shifted up 3 places (to put
M = 0). If c is not or 1 there is a loop stop.

In state 5 numbers are read in. As each digit is read in, the partly formed number is multiplied
by 10 and the new digit is added. The character © indicates a fraction; it initiates a process whereby
a power of 10 is built up as the succeeding digits are read (this starts as 1 and is multiplied by 10 as
each digit Is found). The end of the number Is shown by Sp or CR LP. Erase is ignored, and other
characters cause a loop stop.

State 6 is also known as the cv- search (alpha- search); In this state Input ignores blank tape (0)
and the Er character only. The characters CR LP or LP cause the /S- search (state 1) to be entered, and
A. causes Input to be left for the Decode section dealing with directives. All other characters cause a
loop stop. Note that it is only in this state that (p is ignored.

The above details are summarized in Table 7.5, which also gives a little extra information. In this
table X represents the partially built up order or number; it is always zero on entry (to state 1 or 2).
The quantity y is the power of 10 which is built up during input of a fraction; it can be taken as zero
initially. The quantity c is the octal digit counter used to make sure that there are either 3 or 4
octal digits after the yv-address of an order. The symbols TO 4' mean enter state 4, and so on. The
expression END (F (BDER means that the following events take place:

o'order: x is now the a-order and is kept; x is set equal to and state 2 is entered.

b-order: x is now the 6-order; the a-order and the stop/go digit are added and the resulting
order- pair is stored in the location specified by the T. A. , which is then increased
by 1. State 1 (/S- search) is then entered (with x = 0).
The phrase END OP NUMBER means that firstly the sign of x is changed if a minus sign was read, secondly
X is divided by y (the power of 10) If a point © was read, and finally the result is stored in the
place specified by the T.A. , which is then increased by 1. We then return to state 1 (/3-search) after
setting * = 0.

If overflow occurs at any stage of the operations of Input It Is not Immediately detected, but
OVR remains set as there are no 23-, 64- or 65-orders anywhere In Input. The computer will consequently
stop when attempting to write the result Into the main store. Such overflow Is most likely to be
caused by one of the following:

(a) two many digits in a number,

(b) number too big,

Note that the Integer -2^^ may not be punched as -274877906944, since the Intermediate result 2^^ will
be built up before the sign is changed at the end. Fractions may not have more than 11 digits after
the point, or the power of 10 (I.e. y) will overflow.. If we punch a number such as +1.2 overflow will
occur when Input divides * by y (I.e. 12 by 10 for this number); the number -1.0, on the other hand,
leads to the permissible division of -10 by 10. Zeros in front of the © in a fraction have no effect.

The y' search (gamma- search) Is not a part of Input, but Is In the Decode sequence. It Is called
In whenever A. Is read by Input in states 1 or 6, and has the function of reading in the next character,
ignoring only Er; this is a warning character and is then dealt with appropriately (after a check
has been made that the next character is 0, again ignoring Er).

The various stops which occur in Input are Included in Tables 7.7 and 7.8 which show the error stops
In the Initial Orders. These stops are powerful aids to the detection of punching errors.

Input Is called In In a special way after an X-directive so as to read in single orders. These
orders are read In as though they were a-orders (e. g. LP is ignored) but a loop stop occurs if a letter
shift or a sign is read In state 1. The action when the order is complete is rather different, of
course.

The Address Input section of the Initial Orders has the function of reading the one or two
addresses punched In a tape directive. Each of these addresses Is stored in the same form: as a
modifier, with the slgn-dlglt of the word Indicating whether the address Is that of an a-order or a
fc-order (the slgn-blt Is 1 In a 6-order address). A single address is left in JfS and in f/5.4; If
there are two addresses the first Is left In J3 and {75.3 and the second In U5.^. Address Input can be
In any of three states at the various stages of reading each address; the states can be numbered 1, 2
and 3. The effects of various characters In each state are summarized In Table 7. 6, In which x
represents the address being built up. The word (STOP) In brackets means that OVR gets set, which
causes a stop when CR LP Is read at the end of the directive.

- 162

THE INITIAL ORDERS

7.8

Tape
character

/3- search

First
character

First

digit of

6-order

Remaining

decimal digits

of orders

Octal digits
of orders

Numbers

a- search
Blank
tape

- 9
(decimal
digit d)

First digit
of a- order,

x' = d
TO 3.

First digit
of fe -order.

x' = d

TO 3.

Next decimal

digit of order.

x' = lOx + d,

TO 3.

Next octal digit of order.

x^ = nx ^ d.

c' =■■ c - 1.

TO 4.

Next digit

of number.

x' = lOx + d,

y' = lOy.

TO 5.

STOP

Mark +
TO 5.

STOP

Add relativizer
to X,

c' - 4.
TO 4.

Retain only 6 l,s. bits
ot X, x' = X 8i, QZ
c' - 4.
TO 4,

STOP

Mark -
TO 5.

STOP

c' = c - 1.
TO 4.

STOP

x' = X + B,

c' = 4.

TO 4.

If c = mark stop
order-pair. TO 4,

Set y = 1

and mark

as fraction,

but STOP if

already so

marked.

TO 5.

STOP

If c ?^ STOP

CR LP

Ignore.
TO 1.

Ignore.
TO 2.

END OP
ORDER

If c = 0, END OP ORDER

END OP
NUMBER

Enter

/3- search
TO 1.

If c = I, x' = Bx then
END OP ORDER

If c ?^ or 1, STOP

Ignore.
TO 1,

STOP

a-- order;

Ignore.

TO 3,

a-order: Ignore.
TO 4,

STOP

Enter

/J-search

TO 1.

6-order:
STOP

6- order: STOP

Ignore.
TO 1.

Ignore.
TO 2.

c' = 4
TO 4.

If c = 0, ignore

> TO 4.

END OF
NUMBER

STOP

If c= 1, jk^^Sat,
c' = 0,

If c ?^ or 1.

c' = 4 J

Ignore,
TO 1.

Ignore.
TO 2.

Ignore.
TO 3.

Ignore.
TO 4.

Ignore.
TO 5.

Ignore.
TO 6.

- A.

Enter
y- search.

STOP

Enter
y- search

Enter

a- search.

TO 6,

STOP

Ignore.
TO 6.

All
others

STOP

Table 7.5 The effects of various characters read by Input.

In a manual directive the address (if required) is taken from the handswitches in the Decode
sequence; Address Input is not called in.

There are many checks in the Initial Orders which are Intended to detect punching errors and, by
use of the odd-parity tape code, to detect malfunctioning of the tape-readers and tape-editing
equipment. The most common punching errors lead to a loop stop in 0. 5+. Table 7.7 shows all the loop
stops in the Initial Orders; some of them concern features which have not yet been described. The
stops are given in order-number sequence to permit identification on the monitor.

163

7.8

THE INITIAL ORDERS

Tape
character

First digit
of address

Remaining decimal

digits of

address

Octal digit
of address

- 9

(decimal
digit d)

x' = d.
TO 2.

x^ = IQx-^ d,
TO 2.

;t ' = X + (f,
TO 3,

(STOP)

x' = 8

{x + relativizer),

TO 3.

If marked as 6-
order address:
(STOP) Otherwise:
mark as ft -order
address: TO 3,

(STOP)

If a second address has been marked:
(STOP). Otherwise: x to 5.3
(first address), mark second address to
come, TO 1.

(STOP)

x' = 8;c, TO 3.

Ignore. TO 3.

LP
Sp
Er

Ignore,
TO 1.

Ignore, TO 2.

Ignore, TO 3.

CR LP

(STOP)

If one address: x to X2 and 5.4 (address)
If two addresses: first to X2 (and 5,3),

X to 5.4 (second address),
Then: If OVR set: STOP.

If OVR clear: Return to Decode.

All
others

STOP

Table 7.6 The effects of various characters in Address Input

Order
number

Order

Cause of stop

0.0

3 63

6-order address after T, orB with negative T.A.

0.5

0.0

7 60 6

Unassigned warning character.

0.5

1 63

t > 4 in sequence \.A<p t (No magnetic tape)

0.5+

5 61

Wrong character, or no after warning character.

to. 6

0.6

1 63

t > 6 in sequence X.A0 t

0.6
0.6+

7 62
2 62

A. in order pair or number or after X or R;
may be due to omitted order.

tl.2+

1.2+

7 60

Q-directive, routine not available.

1.5

5 61

CR not followed by LP after warning character address.

1.6

2 62

Sign (+ or -) at start of fe-order or after X or R.

2.2+
2.3

-^ 7
_2. 2+

1 01

4 67

•

Binary input. Order overflow due to addition
of Relative Address (9'/2 hr. stop)

2.6

5 61

Directive address overflowing or wrongly punched.

t3.1

3.1

1 63

Assembly cue-list address too small.

3.3

3 63

P. G, I or S followed by 6-order addresses.

3.4

6 61

Binary input. Checksum disagreement.

tt3.5

3.5

1 61

Assembly. L and (^s after cue list not followed
by kfi or CR LP.

4.1

4. 1

3 63

Either (a) T followed by 6-order address or
(b) B read with C(5.7) negative.

t4.2-4.3+

Loop after handswitch Q, if optional stops inhibited
(see table 7. 10, order number 4. 2). Clear handswitches,
operate Run key, and tap out the routine number.

Table 7.7 Loop stops in the Initial Orders

t Not applicable to 4096 store.

tt In the 4096 Assembly this error causes a loop stop in 3.4.
- 164 -

THE

INITIAL ORDERS

7.8

There are four places In the Initial Orders where a writing-with-overflow stop may occur; these
are shown in Table 7.8. It should be noted that if a 71- or 73-order is encountered with OVR set the
computer stops instead of writing into the main store, and it stops as if it were about to obey the
next order. This next order is consequently shown in the table and may help to identify the stop on
the monitor. The states of the order-number register and the "next-order" lights when the stop occurs
are given in the first column.

Order
number

Orders

Reason for overflow

1.1 (6)

(

4 73 5)

Binary input. Relative Address too large

2.3

4 67

after binary T.

1.2 (a)

(

71 3)

After X-directive. New order overflowing.

1.7

5 63

tl.3 (b)

( 20

7 11 )

furious information in block 0, section 0,

3 76

of a magnetic tape.

3.0 (b)

(

71 6)

Input. Tape word overflowing; may be due

2 02

to, for example: -

(a) too many digits in a number

(b) number too big

Table 7.8 Writing' with- over flow stops in the Initial Orders

Order

number

to.

(b)

to. 5

(6)

0.6

(b)

to. 7

(b)

4.0

(a)

4.0

(b)

t4.1

(b)

4.5

(a)

Cause of stop and usual action required

Magnetic tape isolation or 16/32 word key wrongly set.

Routine read by a handswltch Q. Operate Run key to enter the Initial Orders.

Z-directlve. Operate Run key after changing tapes (or handswitch setting).

Magnetic tape library not isolated or not mounted on the selected mechanism
after Q-directive.

E-directive. Operate Run key to enter programme; entry address is in 3^,^.

Manual X-directive. Set new order on handswitches and operate Run key.

Magnetic tape library not isolated or not mounted on the selected mechanism.

Most manual directives. Set new manual directive and operate Run key for
restart.

Table 7.9 77'Stops in the Initial Orders

The 77-stops and optional stops in the Initial Orders are listed in Tables 7.9 and 7. 10. In the
first column of these tables are given the order-number, as shown on the monitor, and the "next-order"
light visible when the stop occurs.

Order
number

Cause of stop and usual action required

0.7 \a)

Y-directive. Operate Run key after changing tapes (or handswitch setting)

3.7 (a)

C-directive. Specified routine not read in (see Sec. 8.8).

t4. 2 (a)

After Q from handswitches, clear handswitches and Run. (There will be

a loop stop in 4.2 to 4.3 if optional stops are inhibited. See table 7.7).

4.6 (a)

Manual P, I, K, P or X. Operate Run key without changing handswitches to

deal with next order or number in the store; or set new manual directive

and operate Run key for restart.

Table 7.10 Optional stops in the Initial Orders

t Not applicable to the 4096-word store.

- 165 -

Chapter 8

Assembly

The building up of a complete programme from its component routines can be a complicated task.
The Assembly section of the Initial Orders is designed to do it automatically and to introduce as muc
flexibility as possible in the writing of subroutines and master-programmes,

8. 1 The purpose of Assembly

A complete programme is usually made up of some or all of the following items.

(a) A master- programme,

(b) Subroutines from the Library,

There may also be lists of numbers or other constants but we shall disregard these for the present. T
subroutines which are not in the Library, and which will probably have been drawn up specially for the
programme, will be referred to as programmer's subroutines (or private subroutines). Generally
speaking, most of the detailed work is done by the subroutines; the task of the master-programme beini
to organize the calculation, by counting, setting numbers and modifiers needed by subroutines, obeying
cues to call in subroutines, dealing with the results, and so on. The master-programme may, of course,
undertake some calculation Itself. The whole question is discussed in Section 4.2.

The Assembly routine is a section of the Initial Orders which assists in building up inside the
computer a complete programme made up of a master-programme and subroutines as described above. One of
the problems in assembling a complete programme is the allocation of space in the main store for each
part of the programme. It is most undesirable to have to fix in advance the address of each subroutine;
and yet the cues in the master-programme (and elsewhere) will depend on these addresses. In fact, using
only the facilities so far described in this book, it is not possible to write the cues in the master-
programme until the whole of this programme has been written; for it is only then that we know exactly
how much storage space it occupies and so can allocate definite addresses to all the subroutines. A
possible technique which we can adopt during the writing of the master-programme is to leave blank any
word where a cue is needed, and simultaneously to make a note in a special list to record which cue Is
needed to which subroutine, and where it is to be written. When the master-programme has been completed
(apart from the cues) we can allocate the main store and fix the address of each subroutine (it is only
at this stage that we can be sure which subroutines are needed). We can then go through the list of cues
wanted and fill in the cues in the blank spaces left for them in the master-programme.

Since this last process is quite automa,tlc and is subject to definite rules we can programme the
computer to do it; it is in fact one of the things done by Assembly. The allocation of the store among
the various parts of the programme is also undertaken by Assembly. This is a straightforward task; the
parts of the programme are simply placed in the store consecutively as they are read in, except that
each part is made to start at the beginning of a block. The routines are stored as though there were
simply a B-directive before each. Assembly will also select the Library subroutines needed by a
programme and incorporate them in the programme. To use Assembly we must punch certain special
directives on our tapes as explained below (see Sec. 8.2). We prepare a programme tape containing these
directives together with the master- programme and the programmer's subroutines, but not including any
Library subroutines: this tape is then placed in the main tape-reader. In the second tape-reaxter we
put the Library Tape, which contains all the subroutines in the Library. After the programme tape has
been read in by the Initial Orders (acting under the control of Assembly) the Library Tape is scanned;
as each subroutine is found it is examined to see whether or not it is needed by the programme. Unwanted
subroutines are rejected but the others are put into the store, each starting at the next available
block. Usually all the subroutines required are found fairly quickly and Assembly then stops scanning
the Library Tape. At this stage the whole of the programme, including all the subroutines, is held In
the Main store but there are blanks here and there where cues are required. Assembly now processes the
programme by putting in all these cues; the programme is then complete and can be entered in the usual
way, for example by an E-directive.

The operations of Assembly thus fall into two parts: the first is the input of the master- programme
and the subroutines (including Library subroutines), and the second is the processing of the programme,
which consists primarily of filling in the cues (which could not be put in earlier). The first part
usually takes a minute or so and the second part less than a second; when this has all been done the
complete programme is ready to be obeyed and we have finished with Assembly altogether.

In addition to dealing with cues, Assembly also provides the facility of preset-parameters. A
parameter is a constant or number which affects in some way the operation of a subroutine, but whose
value cannot be specified at the time when the subroutine is written. A programme -parameter is a
parameter which is set (i.e. given a particular value) in the programme; usually it is set by the
master- programme putting some number into an accumulator just before obeying the cue to call in the
subroutine (see Sec. 4.2). A preset-parameter Is given its value as a part of the input process of the
programme, and thereafter remains fixed, whereas a programme-parameter can be given several values during

- 166

Plate 1: A general view of Pegasus 2

Plate 2: The control desk of Pegasus 1

Plate 3: The paper tape input equipment

Plate 4: The paper tape output equipment of Pegasus 2

Plate 5: A tape box type A4 with punched paper tape

Plate 6: A hand spooler type A13

Plate 7: The full set of tape-editing equipment

Plate 8: The simplified set of tape-editing equipment

iff f;|;'i- ' '■"''

Plate 9: A keyboard perforator

Plate 10: The keyboard of a Creed Model 75 teleprinter

Plate 11: The programmers' control panel of Pegasus 2

Plate 12: The monitor panel of Pegasus 2

Plate 13: A unlpunch in use for tape splicing

Plate 14: The controls of an ElectroData magnetic tape mechanism

Plate 15: ElectroData magnetic tape mechanisms attached to Pegasus 2

Plate 16: A card reader (left) and punch (right) attached to Pegasus 2

ASSEMBLY 8.2

the course of the programme. A preset-parameter is a word which has to be incorporated in a subroutine
during input of the programme; it will be punched elsewhere on the tape, normally with the master-
programme. The preset- parameters are eventually put into their subroutines during the Assembly
processing. The kind of parameter appropriate to a particular subroutine is something which is
decided when the subroutine is written; instructions concerning parameters appear in the specification
of the subroutine.

For the purposes of Assembly the subroutines in a programme must be numbered, and no two pf them
may bear the same number. This routine number, as it is called, is used to identify the subroutine; it
is simply an integer between and 1023 and is normally given in the specification preceded by a letter
R. Thus, for example, we talk of R 240 when we mean routine number 240. Routine numbers less than
1000 are permanently allocated to Library routines according to a plan given in Appendix 7. Routine
numbers between 1000 and 1023 are conventionally used for programmer's subroutines, these numbers are
used only temporarily and have no permanent significance outside any particular programme. Routine
numbers greater than 1023 are sometimes allocated to programmes for convenience of reference; they
cannot be used with Assembly.

It is not absolutely necessary to use Assembly; indeed in earlier chapters we have described
some simple complete programmes which were put together "by hand". The main Library subroutines are,
however, built up round the facilities which Assembly provides and we must use Assembly if we require
any of them. Many programmes are split up into a fair number of subroutines and the labour of manually
inserting all the cross-references would be very heavy, whereas we can use Assembly to do the whole
process automatically. Assembly also makes it simple to alter a programme; for example we can replace
incorrect versions of programmer' s subroutines by new versions which need not occupy the same amount
of storage space; or we can change the values of preset-parameters. In fact the programme is flexible.

8. 2 Cues and tags

A cue is normally an order-pair for calling in a subroutine. The cue is obeyed in the master-
programme (or perhaps in another subroutine), but we cannot write it there until we know where the
subroutine is stored. To take a simple example, suppose a subroutine is so written that it is to be
entered by bringing its first block into 170 and jumping to 0.2. If this subroutine is stored in B12
onwards (i.e. the address of the subroutine is B12, 0) then we must write the following cue in the master
programme to call it in:

12 072

0.2 60

But if the address of the subroutine is B18.0 the first order of the cue has to be changed to 18 [o]72.
In general we can write the cue in the relative form

0+[o]72

0.2 60

provided it is understood that the relative address refers to the subroutine and not to the master-
programme. We may not write or punch the cue in this form in the master-programme, because, if we did,
the relativizer when the cue was read from the tape would then be that of the master-programme. If,
however, the cue is punched (in its relative form) together with the subroutine then the correct
relativizer will be added to it. This is what is done when Assembly is used.

In the master- programme we leave a gap for the cue by writing the integer +0 (which occupies a
whole word) instead of the cue order-pair. In addition we make a note (a call for a cue) in a special
list, indicating which subroutine is to be called in by the cue (and also which of the cues is required,
since many subroutines will have more than one possible cue each). Along with each subroutine is
punched its cue (or cues) in relative form, and Assembly later picks out the appropriate cue and puts
it in the gap left in the master-programme. In fact the cues get added in but this makes no difference
as a rule, since they get added to the zero stored where the cue is wanted.

The cue or cues for a particular subroutine are punched in the form of a cue- list accompanying the
subroutine on the tape. The various cues are numbered 01, 02, 03, etc... in the order in which they
appear in the cue-list, t The specification of a Library subroutine shows what each particular cue is
used for. The following are two abridged specifications of Library subroutines.

R 5 Double- length number print.

Prints (pq) as a number whose integral part and fractional part are in ^6 and 7 respectively.

Name: D.L. PRINT Store: 7 blocks

Uses: UO; BQ

Cues: 01 to precede the number by CR LP
02 to precede the number by Sp

Link: In 11, obeyed in 0.5

Programme-parameters: The number of digits to be printed before the point (n^) and after it (n2)
are to be set in 4^ and 5^; respectively. If nj^ = the fraction in XI is printed; if n2 =
the integer in X6 is printed (in either case the content of the other accumulator is ignored).

t The numbering system for the cues is octal, so that the cue following cue 07 is cue 10, the next is
cue 11, and so on. This usually does not matter since few subroutines have more than two or three
cues. In fact 08 and 09 have the same effect as 10 and 11 respectively.

167 -

8.2

R 200 Square root, p' = /(pq) (fractions).

ASSEMBLY

Puts into X6 the square root of the double- length fraction in X6 and 7.
Name: SQ.ROOT' Store: 1 block

Uses: f/0; J5, 6, 7

Cue:

Link: In XI, obeyed in 0.7

Error: Not greater than 2"^' (i.e. Vze)

Note: There is a loop stop if (pq) is negative.

It should be noted that the actual cues themselves are not specified; we do not need to know
what they are, only what they cause the subroutine to do. If, for example, we use cue 02 to R5 then
the number will be printed preceded by a space. We can do this by writing +0 in the master-programme
in the place where we want the cue; we must also call for cue 02 to R 5 and specify where we want it to
be added in by Assembly. This is done by a tag.

A tag is a pseudo order-pair preceded by the warning character R (for reference). Various kinds
of tag are used by Assembly, but for the present we consider only togs calling for cues. Along with
our master-programme we punch a number of tags calling for cues, one tag for each of the cues we want.
Suppose, for example, that in B1+. 4 of the master-programme (i.e. in the word in position 4 of block 1+)
we want cue 02 to J15. In this word we write +0 instead of the cue; and we must punch, at the head of
the master- programme, a tag calling for cue 02 to r's to be added into B1+.4. This tag is written as
follows.

R 1 4 -0 2
5 - 01 -

In this tag we write, in the N and X parts of the "a-order", the address (here 1+.4) of the place where
the cue is wanted, just as though it were an absolute address (the relative block-number is in the
;V-address and the position-number in the J-address), After this comes the number of the cue (here 02)
preceded by a minus sign; usually the first digit of the cue number is so that we write -0 in the P
part of the a-order and the second digit of the cue number in the M part. In the 6-order of the tag we
put the routine number (5 in this tag) in the N column. The rest of the fe-order is 01 in the F part
with minus signs on either side of it (in the X and M parts); this 01 is called the function of the tag
and shows that this is a tag calling for a cue. This tag serves only to give information to Assembly;
it is never obeyed as an order-pair.

In general a tag calling for cue number « v to the subroutine with routine number r is written thus.

Address

where

number of

cue

wanted

cue

-'"■^ N

R b

-u

routine/

number

unction of

function of tag (call for cue)

The cue is to be added into the word in position p of block 6+ of the master-programme; this
word is called the tagged word, and since it is normally +0 it will in effect become the cue when
Assembly has processed the programme. As a rule there will be quite a number of these tags punched
in front of the master-programme proper; there will be one tag for every place where a cue is required
(several tags may refer to the same cue, of course, if this cue is wanted in several places). These
tags may be punched in any order. As they are read in they are automatically stored directly in
Assembly's working space in the high-numbered locations in the main store; they are not stored with
the rest of the programme. In fact as soon as a tag has been read in it has the Transfer Address added
to it (in the modifier-position) and the resulting word is then stored in the place specified by the
Reference Address, instead of the Transfer Address (which is left unaltered). The Reference Address
(abbreviated to R.A, ) is set initially to B882.7 and is decreased by 1 every time a tag is read. +
Thus the tags are stored backwards in the upper reaches of the main store without disturbing in any
way the programme they accompany, which gets stored (forwards) in the lower end of the store as usual.
The Reference Address is held during input of the programme in the modifier part of f/5. 5 and can be
inspected there on the monitor tube. The fact that the T.A. gets added into each tag has the effect
of converting the written address (b.p) into the absolute address of the tagged word, i.e. the address
of the word where the cue is wanted. This is because the T.A. remains constant while the tags are
read; and, since they are all punched before the master- pro gramme, the T.A. will be the address of
the beginning of the master-programme.

The directives Al, A2 and A3 play an important role when we use Assembly; L-directives also have
a special use. The Al* directive is punched at the beginning of the programme tape and signifies that
we intend to use Assembly. This directive must be punched before the first tag, and is normally put
just after the D and N at the head of the tape. Next will follow all the tags calling for cues, and
then the master- programme, which must be followed by an L-directive. Assuming for the moment that we
need only Library subroutines then the next item on the tape is an A2'directive, which causes the

t The Reference Address is set initially to B506.0 by the 4096-word store Assembly.

- 168

ASSEMBLY 8.2

Initial Orders to start reading the Library tape in the second tape-reader. The next item on the main
programme tape is an A3-directive, which is read when all the Library subroutines needed have been found
and stored; this directive causes Assembly to process the programme, i.e. to examine all the tags and
insert the cues where required in the master-programme. The last item on the tape is normally an
E-directive such as E 2. to enter the programme. The make-up of the programme tape is therefore as
follows.

D 4 date and serial number

N 4 print name of programme

name of programme

Al 4 prepare to use Assembly

tags calling for cues
master-programme

L "t end of master- programme

A2 4 read the Library tape

A3 "^ process the programme

E 2. -4 — start obeying the programme

To illustrate this we shall now describe a complete programme using Assembly. This programme
tabulates the square roots of all the integers from 1 to 100; it uses R5to print the integers and
the square roots, and R 200 to evaluate the square roots. Abridged specifications of these two Library
subroutines are given above. The master-programme is given in full below.

The integer whose square root is currently being evaluated is denoted by n and is held in X2. The
output of the programme consists of two columns of figures, the first giving n and the second vn; the
lines are grouped into blocks of five lines, and a counter is kept in X3 for this. The square root
subroutine R200 evaluates the square root of the double- length fraction (pq). We use it by setting
this fraction equal to n/256, which is always in range; the answer is then (/")/16 and is multiplied by
16 before printing.

In the master- programme there are three places where cues are wanted (0+. 5, 1+. and 1+. 4). The
integer +0 is written in each of these places and there are three tags calling for cues at the head of
the master-programme. All the necessary directives are included.

All this programme is punched on one tape. The course of events when the programme is run is as
follows. First we put the beginning of the programme tape in the main tape-reader and the beginning of
the Library tape in the second tape-reader and do a Normal Start operation. The T. A. , relativizer and
R. A. are set to their starting values and the computer starts to read the programme tape. The D-directive
causes the date and serial number to be printed and the N-directive causes the name to be printed. Next
the Al-directive is read and there is optional printing of Al; the first part of Assembly is then
entered, which causes a few counters and modifiers to be set in Assembly's working space (actually in
B895t) and a special link to be placed in BO, 1 (ready for subsequent L-directives). The Reference
Address is then set to its starting value of 882.7, but with this programme this operation is actually
redundant. Next a B-operation occurs, i.e. the T. A. is set to the beginning of a block and the relativizer
is put equal to the block-number. Assembly then calls in the rest of the Initial Orders as a subroutine
to read the tape (every time an L is read Assembly is re-entered). The next items on the tape are the
three tags calling for cues; the relative addresses in these tags are converted into absolute ones
by having the Transfer Address (here 2.0) added to them (note that the relativizer is not used for this).
The tags are stored in J5882. 7, 882.6 and 882.5 respectively, and the T. A. is left untouched at 2.0. Then
the master-programme is read in and placed in B2, B3 and a part of 54; the L-directive at the end causes
Assembly's special link (in BO. 1) to be obeyed and Assembly is re-entered. At this stage Assembly
examines each of the tags to determine which subroutines are required; it builds up an Index (starting
at B883.0) containing one word for each subroutine, called for++ * Next there is a B-operation, which
here has the effect of moving the T. A. forward to 5.0 and setting the relativizer equal to 5 (there
is no optional printing), and Input is called in again to read more tape.

The next item on the tape is the A2-directive, which has the effect of changing over to the second
tape-reader, where the Library tape is waiting. This is the only effect of an A2-directive, apart from
optional printing of A2. At the beginning of the Library tape is an N-directive to identify the tape.
After this comes the subroutines, each preceded by its cue-list, which includes a certain tag Identifying
the subroutine. By inspecting this tag and searching through its index Assembly can determine whether
or not the subroutine is one of those which were called for in the programme. If the subroutine is not
wanted it is rejected and Assembly passes on to the next subroutine on the tape. When a wanted sub-
routine is found Assembly first copies its cue- list up into the top end of the main store (among the
tags) and then reads the subroutine in. In this programme the first accepted subroutine to be read from
the Library tape gets put into B5. 0. Assembly then scans its index to determine whether any more
subroutines are required and, if so, it does a B-operation and returns to read in more of the Library
tape. Ultimately all the wanted subroutines will have been found and Assembly then returns to the main
tape-reader (after a B-operation); the rest of the Library tape is not read.

The next item to be read is the A3-directive on the programme tape, which causes the second part of
Assembly to be entered after optional printing of A3. At this stage the master-programme and all the
subroutines have been read in. Assembly now examines every tag and uses certain information built up in
its index during input. When each call for a cue is found the appropriate cue is extracted from the

t B511 in the 4096-word store.

tt In the 4096-word store, the Reference Address is set to a starting value of 506.7, and the Assembly
index begins in B507.0.

# The tags are also changed slightly, the routine number being replaced by the address of the index-word.

- 169 -

8.2

ASSEMBLY

D
N
TABULATE SQUARE ROOTS OP n = 1(1)100

-0

R 1

-0

200

R 1

-0

Prepare to use Assembly

call for cue 01

to R 5, wanted in 0+. 5

call for cue 01 to R 200,
" wanted in 1+.

call for cue 02 to R5,
wanted in 1+. 4

E 2.0

o:^

0.6

5
'•« —

[0]

0.7

' ••

(30)

1.1

print LP LP

set n - 1 in J2

set block-counter in X3

_^1

1 f (P<1)m - n

prograjnme
parameters

set link
+cue 01 to R5

link

Print
CR LP
and
value
of n

(pq)jr = n.2"8 = n/256
set link

1.0

1.2

1.5

1.7+

1.6

■■®

+cue 01 to R200
16 to XI

(P^)m = ^n

programme
parameters

set link

+cue 02 to R 5

16^"
to X6

count lines
in block

link

Print
space
' and
value

of Vn

Print LP and reset
" block-counter at
end of block

n-lOO

stop after r

2.0

= 100

n' = n + 1 '

Repeat
wi th

restore W)

► next
value

0.2+

of n

L End of master-programme

A2 Read Library tape

A3 Process the programme

E 2.0 Obey the programme

cue list for the specified subroutine and added into the tagged word in the master-programme (overflow
is disregarded in this process). In this way all the cues get put in. When all this processing is
complete there is some optional printing which shows where the various subroutines have been put and
gives a little extra information. There is then a B-operation and more tape is read in. The next
item on the tape is the directive E 2. 0, which causes the programme to be entered in the usual way.

It will be noted that Assembly always does a B-operation whenever it calls in Input to read
more tape. There is consequently no need to use any B-directives when we are using Assembly.

The main advantages which result from using Assembly in the way just described are as follows.

(a) We do not need to select the tapes for Library subroutines or to feed them into the computer
individually, or to join them on to our own tapes.

(b) We do not have to specify in advance (or, indeed, at all) where the subroutines are to be stored.

- 170

ASSEMBLY

8.3

There are other advantages in using Assembly, which will become clear later.

The subroutines will all be stored head-to-tail after the master- programme, each one beginning a
new block. It is easy to find the total amount of storage space occupied by the programme, when it has
been written, by simply adding up the requirements for each subroutine as given in its specification.
(If this is too much trouble we can always punch a B-directive after the A3 on the programme tape; this will
will cause optional printing of the final value of the T.A. , i.e. the address of the start of the next
free block after the programme.) The Library subroutines are all written on the assumption that they
are placed in the first 128 blocks of the store; if an attempt is made to place any block of such a
subroutine beyond B121 then a relative address in the subroutine will overflow.

8.3 Cue- lists and programmer's subroutines

The subdivision of a complete programme into a master- programme and subroutines introduces a great
amount of flexibility and makes it possible for several people to work simultaneously on different
parts of the programme or, in general, for one part to be written without close attention to the other
parts. We shall now describe how a subroutine may be drawn up to use the facilities of Assembly. We
assume for the present that it is a programmer's subroutine, i.e. one required in some particular
programme and not likely to be needed elsewhere. In the programme it will consequently be given a
routine number between 1000 and 1023.

We first of all write the subroutine proper, i.e. the orders and constants which will ultimately
go into the programme. This is done in exactly the same way as if Assembly were not going to be used,
but all references to the main store must be relative; in other words we number the blocks of the
subroutine itself 0+, 1+, etc. We disregard at present the possibility of using preset-parameters,
which does affect the way the subroutine is written; this subject is discussed in Sec. 8.4. When the
subroutine itself has been prepared we can specify what its cues are (in relative form), and we can
draw up the cue-list for the subroutine. This is simply a list of all the possible cues arranged in
some convenient order, which must be adhered to throughout. The cues themselves are numbered 01, 02,
03, ,.., as described above, in the order in which they are written. Suppose the last cue is numbered
03 and the subroutine has been allotted the routine number R 1009, then the following tag, called the
title of the cue'list must be written before the first cue.

R
1009

-0
08

This tag should be compared with those given earlier, which were tags calling for cues. The function of
this tag is 08 (written in the F part of the 6-order) and indicates to Assembly that this is the title
of a cue-list. The routine number is written, as before, in the N part of the fc-order, and the number
of the last cue (i.e. the number of cues, in octal) is written in the F and M parts of the a-order. The
address part of the tag is always zero (in the A^ and X parts of the a-order); minus signs are written
in the same position as before.

In general the title of the cue-list containing 8u + v cues for routine number r is written thus.

number of cues (in octal)

R
r

rout
num

iney

ber

-fl
(t

notion of tag
itle of cue-list)

This tag is punched immediately before the first cue in the cue-list. In practice most subroutines
have only a few cues and we seldom have to specify any octal number of cues greater than say 04, in
fact many subroutines have only one cue. In principle the number of cues may be as great as 63 (i.e.
77 in octal) but this is of rather academic interest.

The tape for the subroutine is made up of the cue-list (headed by its title), which is followed
by an L-directive. Then comes an N-directivet and the name of the subroutine (if required). This is
followed by the subroutine proper and another L-directive, thus:

Title of cue-list

Cue-list

L N

Name of subroutine

Subroutine proper

We can illustrate this by an example of a programmer's subroutine to evaluate a simple function.
At a certain stage in a programme suppose we wish to evaluate one of the functions

t This, if present, must be separated from the L-directive preceding it by blank tape only,
is allowed. This is discussed below and in Sec 8.5.

no CR LF

- 171 -

8.3

ASSEMBLY

fix) ■■

. x^ + 0.321x + 0. 113

X + 0.141

g(x) ■■

x^ - 0.327X + 0.113

X + 0.141

Since these functions are very similar we can write a single subroutine which will evaluate either of
them. Here a: is a positive fraction, less than 'A, in X6 and the function is to be left in 16 by the
subroutine. There are two entry-points to the subroutine, each with its own cue, so that we' may evaluate
either f(x) or g(x) as we please. The subroutine may be written and punched as follows.

-0

1003

0.3

0.7

Title, indicating
two cues to R 1003.

Cue 01, to
^ evaluate fix).

Cue 02, to
evaluate gix).

L N
FUNCTIONS P AND G

Cue- list
(followed by L)

r Name of routine

01-

0.0

02-

+. 141

+. 327

+. 113

0.1

5 00

5 01

0.0

6 11

6 20

0.2

6 01

- —

0.0

6 25

6 00

- —

0.7

1 10

/o.

5 02

\ 0.3-^

> Coefficients

+.327 to X5

X + .321 to X5

x + .Ul to 0.0

;k2 + .327* to Xe and 7

add .113

divide by jc + , 141

quotient to X6

plant and obey link

-.327 to X5

Subroutine proper
(followed by L)

Note that the cues are written in the cue-list in relative form, the addresses in them get converted
into the correct absolute form when they are read in because the relativizer will be added to them. If
the master-programme obeys cue 01 to this subroutine then x in X6 will be replaced by f(x) and the
link in XI is obeyed; if cue 02 is used g(x) will be evaluated instead. These cues can be obtained
by writing the appropriate tags calling for them at the head of the master- programme.

Both the cue- list and the subroutine proper must be terminated by L-directives. The reason for
this should become clear when we consider what happens when this subroutine is read in, which we shall
do shortly. We shall first describe how the programme tape is made up for a programme consisting of

(a) a master-programme,

(b) the above programmer' s subroutine,

- 172 -

i' Programmer' s subroutine

ASSEMBLY 8.3

We write the master-programme in the way described in Sec. 8.2, writing +0 in place of each cue. The
tags calling for cues at the head of the master-programme will now include a few referring to R 1003,
i.e. to the above subroutine, as well as those referring to Library subroutines. The whole programme
tape is made up as before, but the programmer's subroutine is included after the master-programme and
before the A2 on the tape. The programme tape may be described as follows.

D
N
Name of programme

Al <i Prepare to use Assembly

Tags calling for cues

Master-prograjTime |>> Master-programme

Title of cue- list

Cue- list

L N

Name of subroutine

Subroutine proper

A2 ■# Read Library tape (if required)

A3 ■^ Process the programme

E 2.0 ^ Enter the programme

If there are several programmer' s subroutines they are each made up in much the same way and are
inserted (in any order) between the master-programme and the A2-directive. If no Library subroutines
are required the A2-di recti ve should be omitted entirely.

When this programme is read in, the sequence of events is as follows. First, the T. A. , relativizer
and R. A. are set to their starting values by a Normal Start operation and the D and N directives cause
their appropriate printing. Next the Al-directive sets the counters and modifiers required to start
Assembly's operations, sets the link in BO. 1 for later L-directives, does a B-operation and returns
control to Input to read in more tape. The tags calling for cues are next read in and placed in the
upper end of the main store as determined by the Reference Address. The master-programme goes, as
usual, into B2 onwards and the L-directive is then read. Here Assembly examines all the tags read
since the Al-directlve and builds up Its indexof the subroutines required. This index will contain an
entry relating to the programmer' s subroutine. Assembly then does a B-operation and returns to read
more tape.

The next item on the tape is the tag which is the title of the cue- list, and as usual this goes
into the place specified by the R. A. , leaving theT. A. untouched. The cues in the cue-list are then
read in; the first of them will occupy the beginning of a block (because of the B-operation at the
end of the master-programme) and the others follow it in the order in which they are punched. The
L-directive at the end of the cue-list is then read. Assembly now inspects the last tag and determines
that it is the title of the cue-list to R 1003; the entry in Assembly's index is thereupon amended so
as to record the fact that the routine has been found. The whole of the cue-list is then copied into a
place determined by the R.A. (and the number of cues in the list), so that it is now up amongst the
tags in the higher- numbered locations of the main store. The T.A. is then put back to the value it
originally had before the cue-list was read in, and Input is called in to read the subroutine (after a
redundant B-operation). The name of the subroutine is printed and the subroutine proper is then put
into the main store; its first few order-pairs will overwrite the cue-list in its original locations;
this does not matter because the cue- list has been safely copied away for later use. When assessing
the amount of storage space occupied by a subroutine we need not allow anything for its cue- list. The
L-directlve at the end of the subroutine causes Assembly to note that input of the subroutine is
finished; a B-operation occurs and more tape is read in. The A2 and A3 directives have the effects
described earlier.

A Library subroutine is punched in the same way as a programmer' s subroutine with only one
important difference; the function part of the tag which is the title of its cue-list is 28 instead of
08. This is because of the slightly different treatment accorded to Library subroutines. Assembly
assumes when it reads a programmer' s subroutine that this subroutine is wanted, and it therefore always
accepts it. A Library subroutine, on the other hand, is accepted only if a call for a cue to it has
been read in. However, Assembly never accepts two routines with the same number. Assembly determines
whether to accept a subroutine or to reject it when the L is read at the end of the cue-list. At this
point Assembly inspects the last tag, which identifies, by the routine number, both the cue-list and
the immediately following subroutine (not yet read in). If the subroutine is to be accepted then the
cue-list is copied away and the subroutine is read in as described abeve. If on the other hand the
subroutine is to be rejected then it is read in but the T.A. , relativizer, and R.A. are reset at the
end of the subroutine to the values they had before the cue- list was read; the subroutine is therefore
overwritten by the next one on the tape.

The name of a subroutine is punched, with its N, immediately after the cue-list and is treated in
a special way. The name is printed only if the subroutine is accepted and if optional printing is
allowed to occur (HO =0). If optional printing is suppressed, or if the subroutine is rejected then
the part of the tape carrying the name is read and ignored by Assembly itself (this is the only occasion
when Assembly reads tape; it otherwise uses the rest of the Initial Orders, chiefly Input).

- 173 -

8.4

ASSEMBLl^

Not infrequently a subroutine may itself require to call in other subroutines, and we then write
tags calling for cues at the head of the subroutine proper. An example should make this clear.
Consider a programmer' s subroutine to evaluate the function

+ /(;

+ y^)

from values of x and y given in XQ and 7 respectively, the result to be left in A^6. We need the Library
subroutine R 200 to evaluate the square root. We first of all write out the subroutine in the
ordinary way but putting +0 instead of the cue to R 200; we then put a tag calling for the cue at
the head of the subroutine and put an L at the end. The construction so far is just like that of the
master- programme. We now write the cue-list and its title at the top, above the tag calling for the
cue to R 200. This is shown in the subi'outine as given below, in which the routine number is R 1002.
This subroutine has only one cue, which is therefore referred to as cue 01; and, as usual, the N-
directive giving the name of the subroutine is punched immediately after the cue-list.

-0

1002

0.0

Title, indicating one
cue to R 1002

cue 01

Cue-list
(followed
by L)

L N

SPECIAL. FUNCTION

Name of subroutine

R
200

-0
01

M-

0.0

-zsi

—

71
00

0.4

0.5

0.6

call for cue 01 to R 200.
wanted in 0+. 3

X to A'4

y to J5
x^

rP- + V'

double-length

Subroutine proper
(followed by L)

plant main link in B0+.6
set link Call in R 200

+ cue 01
to R 200

link

vV+y2 to J6

+ /a;2 +

restore main link in X\

As many tags calling for cues as are needed may be written (in any order) at the head of the
subroutine proper, as shown above. When the subroutine tape Is read in by Input, under the control of
Assembly, the tag calling for the cue to R 200 (and any other similar tags) will be stored in the place
specified by the Reference Address, and not with the subroutine. When the L at the end of the subroutine
is read. Assembly examines each of these tags and extends, if necessary, its index of wanted subroutines.
When the above subroutine is read, for example, Assembly will record the fact that R 200 is now needed
in the programme, even if it was not referred to by any tags associated with the master-programme.
During the processing of the programme caused by the A3-directive the cue to R 200 will be inserted in
the programmer' s subroutine.

8.4 Preset-parameters

Many subroutines can be made more versatile if they are made to use parameters. The detailed
operation of a subroutine can be made to depend on some constant or number, whose value is not fixed
when the subroutine is written. Parameters are of particular value in the construction of a Library
of subroutines because the increased flexibility they provide enables the total number of subroutines

- 174

ASSEMBLY

8.4

needed to be reduced; a single subroutine can do the work of several. A programmer' s subroutine is
written for a particular programme and there is sometimes less need to make it readily adaptable. The
subject of programme-parameters has already been discussed (in Sec. 4.2); these are usually integers
placed in accumulators by the master- programme just before obeying the cue to the subroutine.
Programme -parameters have the advantage that they can be readily changed during the course of the
programme. Sometimes this advantage is not wanted and it then becomes a nuisance to have to set the
parameters every time the subroutine is called in. Consequently some subroutines are written to use
preset-parameters, whose values are fixed for any particular programme but may be changed from one
programme to another. We shall first describe how preset-parameters are used with Library subroutines.

The preset- parameters (if any) for a particular subroutine are grouped together in the form of a
parameter-list, at the head of which is punched a special tag called the title of the parameter^list.
The specification of any Library subroutine requiring such a parameter- list always gives this tag In
full and explains how the parameter- list is to be drawn up. The parameter- list and its title are
normally punched as a, part of the master-programme, but they may appear elsewhere if convenient.

Per example, there is in the Library a subroutine for the evaluation of natural logarithms which
has the routine number R 224. This subroutine works out a scaled down value of the logarithm of the
double-length fraction (pq). This fraction must be positive and is, of course, less than 1; its
logarithm is consequently negative. If the fraction is very small its logarithm is large and negative
(e.g. if (pq) = 1/20 the logarithm is about -3; if (pq) = 1/400 the logarithm is about -6). To
prevent overflow the subroutine consequently evaluates

2~" logg (pq),

i.e. the logarithm divided by 2", where n is chosen to prevent overflow. If we make n large there is
no possibility of overflow but the logarithm is scaled down heavily and precision will be lost if (pq)
is not small. If we know that (pq) is not going to be small in a particular programme then we should
choose as small a value of n as possible. The subroutine has consequently been written so that n is
determined by a parameter- list (containing one preset-parameter); and the specification t gives this
parameter- list as follows.

-0

224

Title, indicating one
parameter for R 224.

Preset-parameter 01

Parameter- list
for R 224

When we wish to use R 224 we simply have to copy this parameter-list into our master-programme, exactly
as it is given here, except that we must of course put in the value of n we require. Note that we do
not put an L at the end of a parameter- list (unless, of course, the parameter- list is at the end of the
master- programme, when it will be followed by the L terminating the master-programme). When the
parameter- list is read in, together with the master-programme, the title, being a tag, will be stored
in the place specified by the R.A. , along with the other tags; but the parameters themselves will
occupy space in the ordinary part of the main store - in other words, they remain where they are put
originally. They are not copied away elsewhere like a cue-list. Note that a parameter-list may be
put anywhere into the main store; there is no need for it to start a new block.

In a parameter- list the preset-parameters are referred to by the numbers 01, 02, 03, etc.. in
the same way as the cues (we sometimes abbreviate "preset-parameter 02" to PP02, and so on). In the
title of the parameter- list the function (in the F part of the "fa-order") is 04; the routine number
is written in the usual place and the number of the last parameter (i.e. the octal number of parameters)
is written in the top right-hand corner. The address in the tag is given as zero above, and In all
titles shown in Library specifications; this means simply that the parameter- list follows immediately
after the tag (the T.A. is added to the written address before the tag is stored).

If desired the tag can be written with all the tags calling for cues at the head of the master-
programme, leaving the parameter- list somewhere later on the tape; if this is done the correct
(relative) address of the first parameter should be written in the address part of the' tag (just as we
write the address of the tagged word in a tag calling for a cue - in fact in a parameter- list the first
preset-parameter is the tagged word).

In the subroutine there are tags calling for parameters, which resemble tags calling for cues,
except that the function used is 02 instead of 01. During input of the programme tape Assembly notes
in the index-word of a subroutine the address of any parameter- list found; it derives this a(idress
from the title of the parameter- list. When the programme is processed (i.e. after the A3-directive has
been read) Assembly deals with a call for a parameter in much the same way as a call for a cue,
extracting the parameter wanted from the appropriate parameter- list and adding it into the tagged word
in the subroutine (disregarding overflow).

In general the tag forming the title of the parameter- list to i-outine number r is written, if the
list contains 8u + v preset-parameters and is immediately preceded by the title, as follows.

R
r

routine number

number of parameters (octal)

function of the tag (title of a parameter- list),

t An abridged specification of R 224 is given on page 181.

175

J. 4

ASSEMBLY

And a tag calling for the addition of preset-parameter number uv^in the parameter- list of routine
number r is written as follows.

address (relative)

number of parameter called for

R 6
r

-u V
02 ~

routine number.

function of the tag (call for a parameter)

The address in this tag is that of the word which is to have the parameter added to it; like all the
addresses in tags it is a relative address. There may be any number of calls for parameters but there
is only one effective parameter- list for a routine. If more than one parameter- list is supplied for
some routine then the last to be read in will be used.

Preset-parameters may be applied just as well to programmer' s subroutines as to Library subroutines.
The parameter- list normally forms part of the master-programme though it may be read in at some other
stage if this is desirable. The tags calling for parameters will normally appear at the head of the
subroutine proper, that is after the cue- list and the name of the subroutine.

To illustrate these points we shall construct a rather trivial programmer' s subroutine to evaluate
the fraction

2-" /(p

2 + q'^).

where n is specified by a preset-parameter, and p and q are integers in 16 and 7 on entry; the result
is to be left in X6. We can suppose that n is left unspecified when the subroutine is written so that
it can be made as small as possible (without overflow occurring) to suit the requirements of the
problem. For example if p and q are both less than 100 in absolute value we can safely set n = 8; but
if p and q are allowed to be as large as 1000 we must set n = 11 to prevent overflow. We shall further
assume that the subroutine is called in "at only one point in the master-programme, so that there is no
need to adopt the usual technique of setting a link in XI; instead, we shall write the subroutine so
that when it has done its work it obeys a preset link, i.e. a link which is built into the subroutine
and is not changed. This link will be another preset- parameter, as we do not wish to specify it before
writing the subroutine. We shall also arrange that, if overflow should occnr during the subroutine
(because n was made too small), then another order-pair is obeyed, which can call in an overflow- routine.
We do this because we do not wish to specify in advance what action is to be taken if overflow should
occur. This order-pair is another preset-parameter (it can be called a cue to the overflow- routine).

The subroutine is given in full below; it has been given the routine nuinber R 1001. The cue-list
contains only one cue and is, as always, terminated by L. Then comes the N-directive including the
name of the routine. This is followed by four tags, one calling for the cue to R 200 to evaluate the
square root, and the other three calling for the parameters to be added to certain words in the
subroutine, which comes next. Preset-parameter 01 is of the following form.

76 -2n

This specifies the value of n; when it is added to the order-pair in 0+. i of the subroutine it
inserts the required shift number in the 54-order so that it will shift up p2 + gS by 76- 2n places.
We have to do this because p and q are integers and the answer is to be a fraction (p'p). Let us write
Pp and qp. for the fractions €p and eg, then

p;. = 2-" v/(p2 + ^2) = v/{2-2» (p2 + ^2)}^
= /2-2"{(238p^)2 + (238 q )2}.

- 176

ASSEMBLY

8.4

-0

1001

El 72

0.0

Title, indicating
one cue to R 1001

, cue 01

L N

SQUARE ROOT FUNCTION

Cue-list
(followed by L)

Name of subroutine

-0

200

-0

1001

-0

1001

-0

1001

ni . "^

7
6
5

___®

0.5
0.4

5
6
5

00
20
22
54
65
00

—

0.0
1

2
3

call for cue 01
to R 200 (in 0+. 3),

call for parameter 01
to R 1001 (in 0+. 1).

call for parameter 02
to RlOOl (in 0+.4).

call for parameter 03
to RlOOl (in 0+.5).

q to X5

p2 + q^.to X6 and 7.

+ PP 01, shift up 76 - 2n places

jump if overflow

set link for R 200

+ cue 01 to R 200

+ PP 02, preset
main link.

+ PP 03. cue to
overflow routine.

Subroutine proper
(followed by L)

The following is a typical parameter- list for this subroutine. The value of n is 8, which determines
preset-parameter 01. When the subroutine has evaluated the required result, it returns to the master-
programme by transferring its block 10+ to f/0 and jumping to 0.3; this is done by preset-parameter 02
(the main link). Preset-parameter 03 is obeyed if overflow occurs in the subroutine; it transfers block
12+ of the master-programme to U2 and jumps to 2.6.

177

8.5

ASSEMBLY

-0

1001

10+

0.3

12+

2.6

Title, indicating

3 parameters for RlOOl

first preset-parameter (n=8)

preset main link

cue to overflow
routine

There are a few points to note here. Preset-parameter 01 is a pseudo order-pair fhich has zeros
everywhere except in the /V- address of the 6-order (its stop/ go digit is because it is marked as a stop
order-pair); when Assembly adds it to the order-pair in 0+. 1 of the subroutine it has the effect of
putting the A^-address in the 6-order. This order-pair will read as follows after the Assembly
processing:

5 5 22

(60) 54

The parameter-list is punched as a part of the master-programme; this means that when it is read in
the relativizer will be that of the master- pro gramme and the parameters 02 and 03 will have the correct
main store addresses in their 72-orders, The overflow routine is simply a pai-t of the master-programme.
Notice how a parameter- list allows us to write and punch a word with the master-programme for later
insertion into a subroutine. A cue, on the other hand, is punched with a subroutine but is later put
into the master- programme. By these two devices Assembly allows in effect cross-references from one
routine to another.

When writing a subroutine we may wish to provide the facility of preset-parameters, and yet we may
be able to supply values for every parameter which will probably be those chosen by the user of the
subroutine. Here an optional parameter-list can be useful. An optional parameter- list is one which is
used by Assembly', only if no other parameter- list is supplied. For example, the logarithm subroutine
R 224, described above, is accompanied on the Library tape by an optional parameter- list which has
n = 5. This fact is mentioned in the specification. If the user of the subroutine does not provide a
parameter- list of his own then Assembly will use the optional list and set n equal to 5. An optional
parameter- list is written In the same way as an ordinary parameter-list except that the tag which is
the title of the optional list has the function 06 instead of 04. Thus the optional parameter- list
supplied with R 224 would be written as follows.

-0

224

Title, indicating one optional
parameter for R 224

Preset-parameter 01 (n = 5)

Note that in an optional parameter-list all the parameters must be given values; it is not possible to
specify values for only a few of them.

Since a parameter-list is always stored along with the programme with which it is punched, allowance
must be made for the storage space occupied by an optional list, whether it is used or not. The
storage space required by a Library subroutine is given in the specification (under the heading Store).
Since the L at the end of the subroutine always causes Assembly to do a B-operation we must allow a
whole number of blocks for the subroutine, including any partially filled blocks occupied by the
optional parameter-list.

8.5 The preparation of a programme for use with Assembly

We shall now collect together some of the information given in earlier sections of this chapter,
add a little new matter and set it all out systematically. We shall also indicate how the directives
and tags used with Assembly are punched.

A general tag for use with Assembly may be written as follows.

R 6
r

-u

178

ASSEMBLY

8.5

This tag is a go pseudo order-pair preceded by a letter R. Its component parts are as follows:

(a) b.p is the relative address of the tagged word,

(b) uv is an octal number (8u + v) ,

(d) ef is the function of the tag.
The various functions (ef) in Assembly tags are as follows;

routine number r:

all these refer to the subroutine with

01 Call for cue number uv,

02 call for preset- parameter number uv,

04 title of a parameter-list containing Su + v parameters,

06 title of an optional parameter- list,

08 title of the cue- list for a programmer' s subroutine,
containing 8u + v cues,

28 title of the cue-list for a Library subroutine.

When the tag is read (by Input) the address has the T.A. added to it; the resulting word is then stored
in the location specified by the Reference Address (R.A. ), which is then reduced by 1; the T.A. is not
changed. The address b.p is therefore an address relative to the present value of the T.A. (and not to
the relativizer). Suppose, for example, that the following tag is read from the tape when the T.A. is
12.4. the relativizer is 2, and the R.A. is 880.3:-

Before the tag is stored the T.A. (12.4) is added to its address (1.7) to give 14.3.
pseudo order-pair is therefore stored in B880.3:~

The following

14 3-0 3
1000 - 01 -

After this has been done the R.A. is reduced to 880.2 but the T.A. is still 12.4. The relativizer plays
no part. This particular tag will cause Assembly to add cue 03 for R 1000 into the word in B14.3, which
is the tagged word. The addresses in the various kinds of tag give the following information to Assembly:

01 to 02 Address of word to which a cue or parameter is to be added.

04 or 06 Address of the first parameter in a parameter-list.

08 or 28 Address of a subroutine (and also of the first cue before the cue-list
is copied away).

The component parts of the tag word are stored as follows:

13 bits

6 bits

3 bits

10 bits

6 bits

b.p + T.A.

ki + V

5e + /

stop/go bit
(always 1)

It is from this stored form of the tag that Assembly derives all its information. During input of the
programme Assembly examines the tags whenever an L is read; the information is then used to build up,
or to extend or amend, an index of subroutines. Each entry in this index is a single word containing
the essential information relating to one subroutine. When the tag is examined the routine number r is
replaced by the address of the corresponding index-word. The index-word contains the routine number,
the address of the cue-list (when copied away), the address of the parameter-list (if any), and a
record of the functions of the tags relating to the subroutine (see Sec. 8.9). The index may not
contain more than 97 entries; consequently no programme may include more than 97 subroutines. '

The usual tags occurring in a master- programme are those calling for cues and those which are
titles of parameter-lists. The tags calling for cues are normally all grouped (in arbitrary order) at
the head of the master-programme, in which case the address in each tag is simply the relative address
of the word which is to have the cue added to ittt. The titles of parameter- lists are usually written
immediately before the first preset-parameter in the list, when the address in the tag should be zero.
Alternatively the title tag may be written at the head of the master-programme with the other tags, when
the address in the tag should be the relative address of the first parameter. An L-directive must be
punched at the end of the master programme.

t The index may contain not more than 34 entries in the 4096-word store.

tt An alternative scheme is to write each tag calling for a cue immediately in front of the place where
the cue is required; if this is done the address part of the tag should be zero.

- 179

8.5 ASSEMBLY

A subroutine is always immediately preceded by its cue-list, which is headed by its title tag (no
other tags are allowed In a cue-list) and followed by an L, The address in the title tag should always
be zero; and the function is 08 in a programmer' s subroutine or 28 in a Library subroutine,. The
subroutine proper is terminated by L; it may contain tags calling for cues or for parameters or the
titles of parameter- lists; these are all written in much the same way as tags in the master- pro gramme.
The name of the subroutine is punched immediately after the L following the cue-list; this name is
printed optionally (if /JO = 0) provided the subroutine is accepted.

The tape for a programme starts with D and N directives and the special directive Al, which calls
in Assembly to do certain preliminary operations, including setting the R. A. to its starting value of
882.7. This. directive always causes a B-operation to occur before more tape is read. It must appear
on the tape before the first tag.

The A2-directive is punched after the last programmer's subroutine, or after the master- programme
if there are no such subroutines; it causes the Library tape to be read in on the second tape- reader,
after optional printing of A2. This directive is not restricted to programmes using Assembly, its only
effect being to switch to the second tape- reader. On a normal programme tape (using Assembly) the A2
can be thought of as representing all necessary Library subroutines.

When the last Library subroutine needed for the programme has been read from the Library tape,
Assembly returns to the main tape-reader where there is normally the A3-directive to initiate the
processing of the programme. Assembly now examines all the tags. If a tag is a call for a cue then
the index-word is used to direct Assembly to the place where the cue-list is stored (among the tags),
and the appropriate cue is extracted and added into the tagged word (overflow being disregarded). A
call for a parameter is treated in much the same way. The titles of parameter- lists are ignored at this
stage as the necessary information (i.e. the address of the parameter-list) is contained in the index
word. The title of a cue-list causes Assembly simply to avoid the following cues. (This examination
of the tags is done in a forward direction through the store i.e. starting with the last tag read in.)
The optional printing associated with the A3- directive is very useful; it is as follows.

(a) As soon as the A3 is read there is optional printing of A3.

(b) When the processing is complete there is a line of printing for each subroutine accepted; this
gives

(i) the routine number,

(ii> the address of the subroutine,

(iii) the address of the cue- list,

(iv) the address of the parameter- list (or an asterisk if there is none).

Thus all essential information about the subroutine is available, and we can, for example, print out the
cue- list if we wish, and also determine just where each subroutine has been put. The parameter- list
address is that of the actual list used in the processing (if several lists are supplied for a given
subroutine only the last one read in is effective; an optional list is used only if no other is
supplied). In this printing the routine number consists always of three digits (except that spaces are
printed instead of left-hand zeros); if there is a programmer's subroutine then the first digit of
its routine number is printed as + (representing 10), for example routine number 1009 is printed as +09.

Assembly ceases to read through the Library tape as soon as it is satisfied. Normally this is
when all the specified subroutines have been found. Strictly speaking. Assembly is satisfied only when
it has found each of the following:

(a) a cue-list (and its subroutine) corresponding to each call for a cue,

(b) a call for a parameter for every parameter- list supplied (other than optional lists),

(d) a call for a cue for each programmer's subroutine supplied.

If only Library subroutines are used then the last is irrelevant. For example. Assembly will never be
satisfied if a non-existent subroutine is called for, or if a parameter- list is not supplied for a
subroutine needing one, or if a parameter- list is supplied for a routine which does not need one. In
such circumstances Assembly will continue reading the Library tape seeking satisfaction in vain, and it
will eventually reach the special End of Library routine which appears, after a Y-directive, at the end
of the tape. This routine is entered as soon as it has been read in (it is a kind of interlude) and
prints out information showing which kinds of tag are missing. If, for example, a non-existent R 199
has been called for in error it will show that a tag with function 08 is missing (this is equivalent to
28); it is at this stage possible to supply any missing information, if it is available, by putting it
into the second tape- reader and operating the Run key.

The whole of the programme (except for the Library subroutines) should be punched before the A2, or
Assembly may not be satisfied on reading the Library tape.

The Library tape has to be scanned as rapidly as possible and it is consequently punched in the
special binary code used by Binary Input (see Sec. 7. 7). This is read at the rate of nearly 5 blocks
per second.

The way in which a programme tape is punched is mostly described in Sec. 6. 5, but there are a few
points concerning Assembly which should be amplified. The Al, A2 and A3 directives are punched in a
straightforward way, for example the Al directive is normally punched as follows:

Blank tape CR LP
;VA</)1 CR LF
blank tape

As usual Er (erase) may appear anywhere except between CR and LF. In fact the single digit after the
A is not an address and need not be followed by CR LP; either of the CR LP's indicated above may
therefore be omitted but this is not recommended. It is important to note that, apart from Er, no
character is allowed between the A and the 1, except for a single 4>. For example, it would be wrong
to put a Sp in here and this would lead to a loop stop on input. The L-directives which appear at the

- 180 -

ASSEMBLY

8.5

end of every routine and cue-list are normally punched simply as

Xl blank tape

without any CR LP.

A tag is simply a pseudo order-pair preceded by an R, and is conseauently punched as kKcf) followed
by the usual punching for pseudo order-pairs. It is customary, however, to make the yv- addresses up to
five characters (i.e. four plus a space) to cater for tags referring to programmer's subroutines (which
have a four-digit routine number). For example, the tag

R 3 4-0 2
456 - 01 -

would usually be punched as follows:

K K(p Sp Sp 3 Sp 4 - 02 CR LP
Sp 456 Sp - 01 - CR LP

Note that the letter R counts as one character.

If we punch in this way a tag having a two-digit block-number (in the iV-address of the a-order)
we get a space between the R and the block-number, which improves the appearance of the print-out.
Great care should be taken not to omit any of the minus signs in punching tags.

The name of a subroutine is punched after the cue-list according to the usual rules for N-directives
but the N itself must not be preceded by CR LP. In fact only blank tape (4>) may appear between the L
at the end of the cue- list and the An introducing the name; no character other than </> is permissible
(not even Er). If there is no name (this would be unusual) then the L should be followed by blank tape
and the CR LP before the first word or tag of the subroutine. Generally speaking, the name of a
subroutine should start with CR LP to ensure that it is printed on a new line; for example if a sub-
routine has the name ABCD it should be punched as follows (the L is that at the end of the cue-list):

L blank tape A,N0 CR LP
k ABCD blank tape CR LP
First word or tag of subroutine

If the subroutine is accepted the following characters of the name are printed optionally:

4) CR LF k ABCD <^

Should the N before the name be preceded by CR LP then the name will always be printed, even if the
routine is rejected. Other unacceptable characters (such as Er) cause a loop stop.

Lists of constants and other data may be dealt with in various ways. The simplest is usually
to compute the total storage space occupied by the programme (this can be confirmed by the optional
printing produced by Assembly or by a B-directive written after the A3). Various sections of the
main store can then be allocated, each starting at a fixed address. This method is usually adequate
but sometimes we may programme the computer to determine where data are to be put and which blocks of
locations are to be used as working space. The final value of the T.A. in 5.7„ may be of use.
Occasionally it is best to use Assembly in an unconventional way, for example oy supplying dummy
subroutines consisting of just a cue-list followed by two L-directives; the cue-list can be used to
provide correct relative addresses for automatic insertion Into the master-programme.

We conclude this section with a description of a complete programme for tabulating the function

z = argsech y

-log.

/ (I - y^)

for y = 0.01 (0.01) 0.99. There will be a master-programme to organise the calculation, and a
programmer' s subroutine to evaluate the function. When writing a programme of this sort it is usually
best to write the programmer' s subroutines first and the master-programme last. The master-programme
will use the Library subroutine R5 to print the values of y and z. The programmer's subroutine may be
numbered R 1000; it will call in two Library subroutines, R 200 for the square roots and R 224 for
the natural logarithms. Abridged specifications of the Library subroutines R5 and R 200 are given on
pages 167 and 168.
The following is an abridged specification of R 224: -

R 224 Logarithm, variable range, p' = 2*" log^ (pq).

Puts into X6 the scaled down natural logarithm of the double- length fraction in A'6 and 7.
positive integer n is fixed by a preset-pai'ameter.

The

Name: LOG MK. 2

Uses: UO, 1; X5. 6. 7.

Time: 42 or 58 msec.

Link: In XI, obeyed in 1.7.

Store: 3 blocks.

Cue:
Error:

01
about 2€.

181

8.6 ASSEMBLY

Notes: Only the left half of the normalized ipg) is used; q must not be negative. If (pq) is
negative or the result overflows there is a loop stop.

Hie value of n should be set by a parameter -list of the following form:

-0 1

224

- 04 -

GO 0.

If no parameter- list is supplied by the programmer, n will be set as 5 by an optional
parameter -list.

The fraction y is always in range. As y increases from 0.01 to 0.99 it can easily be found that z
decreases steadily from about 5,3 down to about 0.14. We therefore evaluate z/8 instead of z. In
evaluating the formula we can adopt a quite straightforward approach since the values y = and y = 1
are excluded, and overflow cannot occur. The logarithm subroutine R 224 evaluates

p' = 2-" log ipq).

where n is fixed by a preset-parameter. We put n = 3 and the result is then -z/8. The programmer's
subroutine R 1000 will be called in from one point only in the master -programme and we shall therefore
write it to obey a preset link when it has done its work. In R 1000 there will consequently be tags
calling for cues to R 200 and R 224, a tag calling for a preset-parameter (the preset link) and a
parameter-list (with its title) for R 224. The subroutine has only one cue in its cue-list; when
this cue is obeyed in the master -programme the subroutine replaces the value of y in J6 by z/8.

The master-programme has to call in R 5 to print y. call in R 1000 to evaluate z/8. call in R 5
again to print z and repeat with the next value of y. It starts with y = 0.01; and this has to be
increased 98 times by 0.01. This could be done by repeated addition, but this is inadvisable in a
binary computer since the value of 0.01 will be held only approximately, and repeated addition would
multiply the error. In this programme we store x = 100 y as an integer; x has to take all integer
values from 1 to 99. this can be done by repeated addition of 1 without any error. The value of y can
be obtained by dividing x by 100 (using a rounded division), and this is not subject to the accumulation
of rounding errors.

The complete programme is as on pages 183, 184 and 185.

8.6 The Library

The Library includes, so far as possible, all subroutines and complete programmes of general use.
The subroutines are included on the Library tape; separate tapes of the complete programmes are
available. The Library is continually expanding as new routines are written and included in it; and
from time to time a new Library tape becomes necessary as a result. Subroutines on the Library tape
are democratically arranged, the most popular coming first; this minimizes the time spent in scanning
the tape looking for wanted routines.

Some subroutines appear two or more times on the tape; this is because they are needed by other
subroutines. For example, R 200 (square root) is a frequently needed subroutine and consequently
appears first on the tape. The routine R 242 (arcsin and arccos) is less often wanted and therefore
appears further down the tape; but it uses R 200 as a subroutine and must therefore be followed by a
copy of R 200 in case the latter was not called for by the master-programme (in this event Assembly
would reject the first copy of R 200 as not wanted, but would accept the second, if R 242 were called
for, since there is a tag in R 242 calling for the cue to R 200). Assembly never accepts two copies
of a routine.

The subroutines in the Library are each Identified by their routine numbers, which are allocated
according to the function or purpose of the routine. Each routine number is less than 1000; the
allocation scheme is given In Appendix 7. Complete programmes are allocated routine numbers 7000t
greater than those of corresponding subroutines; such routine numbers are used only for reference
purposes, and cannot be used with Assembly.

Various documents are available concerning the Library. The index to the Library gives brief
particulars of each routine, so as to enable a programmer to see which routines are available to meet
any particular need. A copy of this index is given in Appendix 7. A printed specification of each
routine is prepared for insertion into loose-leaf volumes. The specifications give full details of
how the routines are used. Printed programme sheets for the more frequently used routines are also
prepared, so as to provide full information in case a programmer wishes to adapt a routine to do a
slightly' different job, or if he wishes to use some of the techniques employed in it. A document is
also prepared from time to time showing which routines are available on the current Library tape.

Certain conventions are adopted in writing and using Library subroutines and their specifications;
these should also be used in programmes and programmer's subroutines, if at all possible. The following
are the main conventions; any departure from them must be indicated explicitly in the specification.

1. The link for a subroutine is to be set in XI before the subroutine is entered.

2. In a specification any registers or storage locations whose contents may be altered by a
routine are stated to be used. Nothing should be assumed about their contents after the
routine has finished.

t Pour«diglt routine numbers beginning with 7 are allocated to complete programmes to be run on the 7168-
word store: complete programmes for the 4096-word store have corresponding numbers beginning with 2.

- 182

ASSEMBLY

8.6

TABULATE ARGSECH

Al (prepare Assembly)

-0

R 1

-0

1000

R 1

-0

► calls for cues

E 2.<

J" (^

0.0

.®

0.5

1*.

0.4

—

1.0

(B2)

► Print I J' LP

set block-counter
a; = 1 in J3

y = VlOO to XI

► Stop after y = 0.99

ni -

n2 - 5

Use R5
to print

set link ^^}^

and y.

+ cue 01
to R 5

The master-programme

183

8.6

ASSEMBLY

1.0

1.1

"®

1.7

2.0

(B3)

Link for R5

y to X6

+ cue 01
to R 1000

Use R 1000
to evaluate
z/B.

iPl)M = 2 = argsech y

"1 = 2
n2 = 8

set link

+ cue 02
to R 5

restore f/O

Use R 5
to print
value of z.

link for R 5

2.0

2.2

2
2

40
10

©;©

2
3

40
41

0.2+

-0 1

1000

04 -

1.3

(M)

count lines in block

print LP

reset block-counter
increase a: by 1
repeat loop

title

Preset main
link

Parameter -

list
for R 1000

End of master -programme

-0 1

1000

08 -

0.1

The programmer ' s subroutine

Title

Cue 01

Cue- list

L N
ARGSECH

End of cue -list

- 184

ASSEMBLY

8.6

-0

200

-0

224

R 1

-0

1000

R 1

-0

224

call for cue 01
to R200 (In 0+.3)

call for cue 01
' to R224 (in 0+.7)

call for parameter 01
' to RIOOO (in 1+.2)

Title of parameter -list for

R224, starting at 1+.3

and containing one parameter

0.0

0.4

0.0

'o\

70
01

• link for R 200

-y to J4
-y2 to X6 and 7
l-y2 to XG and 7
set link

+ cue 01
to R 200

/(l-y2)
to X6

+ cue 01 to R 224

'z/8 to X6.

Exit ^

0.0

0.1

L
A2
A3
E 2.0

link for R 224

z/8 to J6

+ preset -parameter 01
= preset main link.

Parameter for R 224.

End of subroutine
Read Library tape
Process the programme
Enter the programme

185 -

8.6 ASSEMBLY

3. A subroutine must be entered in one of the specified ways (normally by a cue) every time it is
used. Only self -preserving subroutines (see below) are left intact in the computing store ready
to be obeyed again. If XI is stated to be used the link will have to be set again, even if the
same one is required.

4. The over flow- indicator (OVR) is to be clear on entry to a Library subroutine and will be left
clear on exit. It is recommended that OVR should be clear at any well-marked division or break
between one part of a programme and another.

5. A subroutine should use as few blocks of ordinary registers as possible. Generally a Library
subroutine will use f/Q and Ul only; some will use only f/0.

6. As few accumulators as possible are to be used. Clearly X6 and XI will be needed by most
subroutines and XI holds the link. If other accumulators are needed they are used in the
order 5. 4. 3, 2.

7. Output routines for the normal 60 char. /sec. punch are usually written to use only UO and to
preserve the accumulators in BO.

8. Any print routine will leave the teleprinter in figure shift, and will assume that it is in
figure shift on entry.

9. Any subroutine requiring a single operand is normally written to take it from X6; if there are
two operands they may be in X6 and XT. If the result is a single number it will normally be
left in X6.

10. Preset-parameters required for addition to order-pairs should normally be positive (i.e. they
will be written as stop order-pairs). This allows the addition of several parameters to a
single order-pair.

11. Programmes and subroutines should be made as simple as possible to use; in particular any
awkward punching of tape or any special and unusual ways of entry should be avoided.

12. All Library subroutines are written on the assumption that they will be stored in the first 128
blocks of the main store (BO to B127). The master -programme and programmer's subroutines should
also be stored here if possible.

These conventions are not arranged in order of importance. A few notes on them may be desirable.
Concerning Nos.l and 9, it is sometimes advantageous to interchange the roles of XI and X6, i.e. to put
the link in X6 and a single operand in XI. The operand can then be the subject of single-word transfers.
Sometimes it may be very difficult to write a subroutine in such a way that it uses f/O and Ul only (see
No. 5 above); it may then be desirable for the subroutine to preserve 172, for example, by obeying some
such order as

4+1373

which copies [/2intoB4+. At the end of the routine the corresponding 72-order can be used to restore
U2. It this technique is adopted f/2 is not included in the blocks of ordinary registers stated to be
used by the routine (nor is BA+ included in the locations used; though of course storage space must be
allowed for it).

A self-preserving subroutine is one which may be entered and re-entered by simple jumps when once
it has been transferred into the computing store. Such a subroutine may either be held entirely in the
computing store and be so arranged that it does not write over any of its own orders, or else it may
arrange to read in its own first few orders again just before obeying the link. Self-preserving
subroutines are often used with computing store links (see Sec. 5.8) and partial cues, which we shall now
describe.

Consider a subroutine which may be transferred into (/3, 4 and 5 and is then self-preserving.
Having once transferred it, the master -programme can repeatedly enter it, for example, by orders like
the following.

(l.3+)l 40 set computing store link
3.0 60

When the subroutine has finished it will obey a pair of orders like these.

25 ) 1 52 shift address to 1

M
60 1 jump to specified address

In this case the subroutine will return to the 6-order in 1.3 of the master-programme. A subroutine
like this is easy to use repeatedly when once it has been transferred into the computing store. To
facilitate the transfers the subroutine is provided with two partial cues (sometimes called 0+ cues),
for example its cues 01 and 02 may be as follows.

► a- order partial cue

02 f 6-order partial cue

The first of these cues contains zeros everywhere except in the A^-address of the a-order, where there is
the block-number of the first block (B0+) of the subroutine. The other cue is similar but has the address

186

ASSEMBLY

8.7

of the subroutine In the /V-address of its 6-order. These cues can be added to 72-orders in the
raaster-programrae so as to make them transfer the subroutine.

Suppose, for example, that the following orders appear in block 4+ of the master -programme.

4+.5

4+.6

By calling for the appropriate cues we can get the address of the subroutine added to the A^-addresses
in these orders. This can be done by writing the following tags calling for cues at the head of the
master -programme (we suppose that the routine number is 1005).

R 4

5-0 2

1005

- 01 -

R 4

6-0 1

1005

- 01 -

R 4

6-0 2

1005

- 01 -

call for cue 02 to R 1005, to be
added into B4+.5

call for cue 01 to R 1005. to be
added into B4+.6.

call for cue 02 to R 1005. to be
' added into B4+.6.

We simply write one tag for each of the orders requiring alteration, a call for cue 01 if it is an
a-order, or for cue 02 if it is a 6-order. Note that the word in B4+.6 is tagged twice; both partial
cues will be added to it. This technique finds other applications; this is the reason why Assembly
adds in the cues and parameters, instead of simply placing them where required.

A few Library subroutines include interludes which do a certain amount of organisational work when
they are read in; these interludes occasionally refer to such things as parameter lists which are
assumed to have been read in previously. It is therefore essential that these lists should appear
before the A2 on the programme tape.

When Assembly has finished the processing of a programme the parameter-lists are normally no
longer needed and may be overwritten. One or two Library subroutines do actually refer to their
parameter-lists (by a table look-up) during their operation. Such lists are properly not preset-
parameters, and they may not of course be overwritten; the specifications of these routines indicate
this fact.

8. 7 The magnetic tape Library!

On an installation with magnetic tape equipment is is possible to read Library routines either
from paper tape, or, more rapidly, from 16-word magnetic tape.

In order to call for subroutines from the magnetic tape Library, the warning characters A5 and A6
are used. A loop steering tape containing this sequence is placed in the second tape-reader in place
of the pajter tape Library, and is read as soon as the A2 on the programme tape is encountered. There
need be no difference in the master programme tape whether the paper tape or magnetic tape Library is
used. Alternatively the A5 may be punched anywhere after the Al on the programme tape, but before the
A6, and the A6 may be punched in place of the A2. The method has the disadvantage that the programme
cannot be so easily used if the magnetic tape Library is not immediately available.

The Kiagnetic tape Library is normally mounted on the highest numbered magnetic tape mechanism
available, as this mechanism is least often required for other purposes. The Al directive places in
B895.1 a tape order to search for the first section of the Library on the highest numbered tape
mechanism. The A5 directive obeys this tape order, and the A6 enters the part of the Initial Orders
which reads the magnetic tape Library. When the required subroutines have been read, control is
returned to the main tape-reader and Assembly initiates a search for the beginning of the magnetic
tape Library.

If it is required to mount the Library on another mechanism the following sequence must be read
after Al and before A5:

X 895,1+

64m (where m is the mechanism number)

The A5 and A6 are normally punched as follows:

Blank tape CR LP

X. A <j6 5 CR LP
A. A 6 CR LP
Blank tape

The punching conventions are the same as for Al. A2 and A3. The single digit after the A is not
an address and need not be followed by CR LP although it is recommended that CR LP should be punched.
Apart from Er, no character other than a single is allowed between the A and the 5 or the A and the 6.

Er may appear anywhere except between CR and LP.

T On the 4096-word store there is no facility in the Initial Orders for reading the Library from magnetic tape.

187

8.8 ASSEMBLY

The warning character Q is used to call for complete Library programmes, such as the Autocode,
which are stored on the same 16-word magnetic tape as the subroutine Library. The Library of complete
programmes, known as the Q-Library, starts one section before the subroutine Library and extends
backwards down the magnetic tape.

The warning character Q followed by the number of the routine required should be punched at the
head of the appropriate tape: this may be a programme tape for an interpretive routine such as the
Autocode or a data tape for a complete programme such as "Simultaneous Equations". When the required
routine has been transferred from magnetic tape to its normal position in the main store the computer
will return to the Initial Orders to read paper tape, except when there is an E or J at the end of the
routine.

Q may be obeyed from the handswitches in the following manner:

(a) Set up Q( 10001) on the last five handswitches in the same way as other warning characters.

The address should normally be zero.

(b) START and RUN.

be a loop in 4.2 and 4.3 until the handswitches are cleared.

(d) Clear the handswitches and operate the Run key to pass the optional stop (if any).

(e) Tap out the routine -number digit by digit, using handswitches to 9 for the digits to 9. As

each digit is tapped it will be punched on the output tape.

(f) If an error is made in tapping the number handswitch 11 should be tapped: this will print

* CR IF X. Q, cp, cancel the number, and return to stage (c) above.

(g) Terminate the routine-number by tapping handswitch 10.

(h) When the required routine has been transferred from magnetic tape to its normal position in
the main store there will be a 77 stop in 0.5 before returning to Initial Orders to consult
the handswitches, except when there is an E or J at the end of the routine.

Most routines are transferred from magnetic tape to a standard position in the main store in the
same way as a binary punched programme tape. When the whole of the selected routine has been transferred
to the main store, the Initial Orders initiates a search for the beginning of the subroutine Library.

If the selected routine includes E- or J- directives these are obeyed in the normal way.

When Q is being read from paper tape or set up on the handswitches, handswitch 14 should normally
be clear. If the Library is mounted on some mechanism other than the highest numbered, handswitch 14
should be down and the mechanism number set up as an octal digit on handswitches 11 to 13. These hand-
switches should be set before operating the Start and Run keys: they may be left set if Q is on paper
tape, but must be cleared before the tapping routine if Q is on the handswitches.

The magnetic tape containing the Library should always be isolated so that no part of the Library
may be accidently overwritten. Every time an A6 or Q is obeyed, a check is made to ensure that the
tape is isolated.

8.8 1*16 C-directive

It is sometimes difficult to make corrections or alterations to subroutines, particularly if the
programme includes many of them. One method is to find out from the Assembly printing where the
particular subroutine has been stored and then to insert corrections by using X- or T-directives. The
short tape effecting the corrections can be inserted near the end of the programme tape, after the A3
but before the E for entering the programme. In this way corrections and alterations can be made to
Library subroutines as well as to those written by the programmer. This method suffers from the
disadvantage of using absolute addresses, so that the corrections may have to be changed if the
subroutines are later stored in different places In the main store. This disadvantage is most serious
when there are many programmer's subroutines, since some of these may well be re-written during the
development of the programme (this may involve moving many of the others) and it may, in any case, be
desirable to test some of them by themselves before the whole programme is put together.

To help overcome these difficulties, the C-directive (C for correction) has been introduced. This
is punched as a letter C followed by a routine-number; it may be used at any time after the subroutine
referred to has been read in, provided that the information stored by Assembly in its working space has
not been spoiled. The effect is simply to set the relatlvizer to the value it had when the specified
routine was being read, i.e. to the address of the first block of the subroutine; the Transfer Address
is not altered. When the relatlvizer has been set in this way, corrections to the specified subroutine
can be made by using directives (such as T. X or S) with relative addresses.

For example, to change the a-order in 2+.3 of the programmer's subroutine R1002 we could punch a
sequence such as the following:

C 1002 Set relatlvizer to start of R1002

X 2+.3

0.7 2 61 new order

This sequence could appear anywhere on the tape after R1002.

This directive can sometimes be used to facilitate cross-references from one routine to another,
for example we can insert a cue to R1002 in 1+.3 of R1005 by some sequence such as the following.

C 1005 set relatlvizer to start of R 1005

T 1+.3 set T.A. to 1+.3 of R 1005

C 1002 set relatlvizer to start of R 1002

0+[0]72
0.1 60

cue to R1002, containing a relative address.

- 188

ASSEMBLY 8.9

This sequence would have to be read in after both of the subroutines referred to.

The optional printing associated with this directive consists of a letter C followed by the routine
number .

If the required subroutine has not been read in when a C-directive is encountered there is an
optional stop in 3.7; if the Run key is then operated the relativizer is set to the beginning of the
isolated store so that most relative corrections will have no effect (the second kind shown above would
then, however, cause a writing-with-overflow stop).

8.9 Detailed description of Assembly

Assembly is divided into two main parts. The first is concerned with everything on the input tapes
up to the A3-directive, it is mainly concerned with the selection of Library subroutines and the
preparation of information to make the processing possible later. The second part is called in by the
A3-directive and inserts all the cues and parameters, and prints (optionally) information concerning the
subroutines accepted. These two parts of Assembly are quite distinct except for an exit sequence, which
is obeyed whenever Assembly returns to read tape. This exit sequence will reset the T.A. and R.A. after
a rejected subroutine; it always does a B-operation and calls in Input.

The first part of Assembly is called in by the Al-dlrective; it then calls in the rest of the
Initial Orders as a subroutine to read in nearly all the programme and Library tapes. These tapes will
be read by Input or Binary Input. Assembly is re-entered whenever an L-directive or its binary
equivalent t is read. As far as Assembly is concerned the tape is therefore regarded as divided into
sections by L-directives. These sections are of three sorts:

(a) Subroutine cue-lists. These are always headed by a title tag with the second function digit 8. and
contain no other tags.

By inspecting the title Assembly decides whether to accept or to reject the subroutine. It will
reject a subroutine only if It has previously accepted the same subroutine (judging by the number) or
if it is a Library subroutine and no other tag has been read with the same routine number. If it is
accepted then the whole cue-list is copied away.

(b) Subroutine programmes. These are always immediately preceded by subroutine cue-lists, and may
contain any tags other than cue-list titles. If the cue-list was accepted then the subroutine programme
is read into the next available locations, except that a new block is always started; these locations
will be the same as those occupied temporarily by the cue-list. If the cue-list was rejected the
subroutine proper is read in as before, but the T.A. and R.A. are then put back to their original values
so that the subroutine will be over-written by the next section of tape. The N-directive which may
appear before the subroutine is treated specially by Assembly,

(c) Other programme. A section of tape which is neither a cue-list nor immediately follows one is
treated as a part of the master -programme and read into the next available locations, starting at a
fresh block.

Assembly uses extensive working space at the upper end of the main store (and BO. 1. where the link
is kept), which may be divided into three parts: tt

(i) Five words in 5895 to hold a tape word, modifiers and counters,
(ii) The index, which occupies space from B883.0 onwards,
(ill) The reference list, made up mostly of tags, which occupies locations working backwards from

B882.7.
The five words in B895 record the following Information: -

895.1 The tape control word planted by Al to search for the first section on the magnetic tape
Library.

895.2 Assembly modifier and index counter:
895. 2j,^ is always 882.0,

895.2c is one more than the number of words in the index.

895.3 Reference Address \ values after the last accepted subroutine or section of

895.4 Transfer Address J master programme, and the B-operation following it.

895.5 After an accepted cue-list and during the following subroutine, 895. 5jw is 1.0 less than
the address of the index-word relating to the subroutine.

The index is compiled by Assembly with one index-word for each subroutine accepted or to be
accepted. Each tag is examined when the L is read at the end of its section of tape; it is then
compared with each index-word. If no index-word has the same routine number a new index-word is
formed, unless the tag is the title of the cue-list of a Library subroutine (its digit 34 is then 1),
when the subroutine is rejected. If an index-word is found with the same routine number as the tag or
if a new index-word has to be formed, then the address of the index-word is written into the tag in
place of the routine number, and the function digits (35 to 38) of the tag are mixed (by an or operation)
with those already in the index-word from previous tags (if any). In this way the last 4 bits in the
index -word contain 0' s in those positions where the corresponding kinds of tag have not yet been
encountered.

If the tag is a call for a cue or parameter nothing further is done to it by the first part of
Assembly. If it is the title of a parameter -list, the address of the parameter- list is written into
the index-word, unless it is an optional parameter-list and another list has been found previously.
Thus a non-optional parameter- list (title tag with function 04) always supplants earlier lists,
whereas an optional parameter -list (title function 06) is only used if it arrives first. The sign of a

t The binary equivalent of an L-directive always has the effect of an Assembly L; it does not in fact
use the word in BO. 1.

T* The 4096-word Assembly uses the following working space:

(i) Four words in B511, 511.2 - 511.5 (which record information corresponding to that held in
895.2 - 895,5 on the 7168-word store).
(11) The index, which occupies space from B507.0 onwards,
(ill) The reference list, made up mostly of tags, which occupies locations working backwards from
B506.7.

189

8.9

ASSEMBLY

parameter -list title is also changed at this stage, so that it becomes a stop tag and is ignored in future
by Assembly (tags whose stop/go digit is are entirely ignored by Assembly).

If the tag is the title of a cue-list, then the cue-list (and the following subroutine) is
rejected if either

(a) the title is that of a Library subroutine and there is no corresponding index-word, or

(b) the corresponding index-word shows that a cue-list has already been found for this subroutine.
If Assembly requires to reject the subroutine programme which follows the cue- list it makes the R.A.
negative (actually -2"^° is put in U5.5). This has the effect that when Assembly is next re-entered
the R.A. and T.A. are set back to what they were before the cue-list was read (these values are stored
in B895, see above). If a rejected subroutine is punched in the ordinary Initial Orders notation, it
is placed in the store as usual but later overwritten, but its tags (e.g. calls for parameters) are
put harmlessly into locations starting at 1023.7 and working backwards; but if it is punched in binary
no storing of programme or tags takes place since the word in (75.5 is negative (Binary Input ignores
everything except for a binary L if C(5.5) is negative; it continues to check the checksums however).

If the cue-list is accepted, it is copied into the reference list with the title preceding it, so
that the last cue occupies the location formerly occupied by the title; and the new address of the
title is entered into the index-word. The number of words copied is determined by the change in T.A.
since Assembly was last left. In addition a word is placed in C895.5 which contains, in the modifier
position, the address (minus one block) of the index-word of the accepted subroutine; this is for
convenience in writing interludes (see below).

If a cue-list is rejected or if optional printing is suppressed (HO = 1), Assembly reads the
N-directive and name (if any) at the head of the subroutine, so that it shall not be printed by the
Initial Orders. To do this Assembly reads and ignores blank tape until it finds A.N or CR LP; /VN
indicates the start of the name and CR LP that there is none, and most other combinations cause a loop
stop.

Whenever the section last read is not a cue-list, the index is examined before returning to read
more tape to see whether every subroutine called for has been found; if so the next tape will be read
from the main tape-reader. If the Library tape was being read this stops, and the main tape next read
in normally starts immediately after the A2.

It will be seen that when the first part of Assembly stops reading the Library tape it has
accumulated enough information in the reference list and the index for the second section to process
the programme as required. This is done, on reading A3, in a more or less straightforward manner by
working forwardst through the reference list up toB882.7.^ Each call tag (with function 01 or 02)
contains the address of the word tagged and the address of the index-word, and the latter contains the
addresses of the cue-list and the parameter-list (if any). On adding parameters and cues to the tagged
words overflow is ignored. At this stage parameter -list titles are ignored (being non-negative) and
cue-list titles cause the following cues to be avoided (the number of them before the next tag is in
the title). When the processing is complete Informative printing occurs (if ^0 = 0). This is done by
working through the index, the subroutine address being obtained from the cue-list title. The T.A.
and R.A, are unaffected by the second part of Assembly, except for a B-operation.

Interludes may be needed for some subroutines. Such an interlude should not be entered by a simple
J-directive such as J a(+) but by a J-directive with two addresses, in fact

J 928.0+ - a(+)

This is because the subroutine may be rejected, in which case an ordinary J-directive would still cause
the interlude to be entered. Other directives are unlikely to have undesirable effects, apart from N.
which is specially treated. The above J-directive causes a sequence starting at B928.0+ to be entered;
if the subroutine is being accepted the second address a(+) is treated as if it were the first, otherwise
Input is immediately re-entered to read the rest of the rejected subroutine. This J-directive must
appear on the tape before the L terminating the subroutine programme. An interlude of this kind is
usually called an optional interlude. Certain information used by Assembly may be useful to such an
interlude, in particular the word in B895.5 (see above) which enables the index -word of the subroutine
to be found. The structure of this index-word and of tags may also be useful; this is shown in the
diagram below. Only 11 bits in the index-word are allotted to the cue-list address, but as this is in
Assembly working space the two most- significant bits are assumed to be ones. All the tags read in will
have been processed by Assembly (when the L was read at the end of their section of tape), except for
those tags, such as calls for parameters, appearing in the subroutine containing the interlude (since the
interlude is entered before the L is read which terminates it).

Unprocessed Assembly Tag:

Digits

1 to 13

14 to 19

20 to 22

23 to 32

34 to 38

Contents'

Tagged
address

Routine
number

Function

t It should be noted that it is only when tags are read by Input or Binary Input that the reference
list is worked through backwards; Assembly always works forwards.

tt For the 4096-word store the -J-directive is J 560.0+ - a( + ).

190 -

ASSEMBLY

8.9

Processed Assembly Tag: t
Digits
Contents*

Index word: **

1 to 13

14 to 19

20. 21

22 to 34

35 to 38

Tagged
address

Index
word
address
-1.0

Function

This digit is zero in a parameter-list title.

Digits

Contents 4

1 to 10

Routine
number

11 to 21

Cue-list

address

768.0

22 to 34

Parameter-*

list

address

35 to 38

Mix of
functions

When an optional interlude has finished its work it should obey the link which was in XI on entry.
There are two alternative links which may be set, depending on whether the subroutine was punched in the
ordinary Initial Orders notation or in binary; care should be taken to obey the actual link set. Either
of these links is self-modified (see Sec. 5.10) and must be in XI at the time it is obeyed. The
interlude should also not disturb the Assembly link stored in BO. 1.

On the 4096-word store, the processed tag Is packed in the same way as on the 7168-word store,
except that the index word address occupies only 12 bits (23-34) and bit 22 is spare.

t On the 4;096-word store, bits 11 to 22 In the index word hold the exact cue-list address and the
parameter- list address is held in the 12 bits 23 to 34. The rest of the index word is the same
as on the 7168-word store.

191

Chapter 9

Some Programming Techniques

In this chapter are described some advanced methods of programming, including the so called
automat ic programming methods.

9. 1 Floating-point operations

In most programmes the problem of scaling arises. The numbers entering into a calculation usually
have to be scaled to bring them in range or to prevent loss of accuracy. In some problems there may
be a physical quantity, for example a pressure denoted by p. Suppose that this pressure p can range
over values from 10 up to 50 Ibs./sq, inch, and we wish to measure it to within 0.1 Ibs./sq. inch.
Inside the computer we can store lOp as an integer and arrange our programme accordingly; alternatively
we can store p/64 or p/100 as a fraction, which is probably to be preferred. Either scheme really
amounts to measuring the pressure in different units (such as lbs. per 100 sq. inches, if we store p as
p/100). The various quantities entering into a calculation must all be scaled in some way unless they
can be conveniently represented directly as integers or fractions. This scaling involves some
rearrangement of the formulae to be programmed. As a rule quantities are scaled by powers of 2 rather
than powers of 10 because we can multiply by a power of 2 (i.e. shift) in any accumulator, whereas we
can multiply directly by a power of 10 only in .^6 and 7. Usually we write a programme in such a way
that all the numerical data for it are punched in their natural, unsealed form, and we use a special
input subroutine to read them in, scale them appropriately and store them. In a similar way the scaled
results of the calculation are 'unsealed' before being printed by a suitable output subroutine. It
should not be forgotten, however, that all the intermediate quantities arising in a calculation, even
if they exist only momentarily, must also be scaled so that they are in range.

In certain problems scaling may be difficult. This may be because the calculations to be done are
very complicated and we may not be able to determine readily the size of all the intermediate quantities.
Or it may happen that certain numbers may vary over a very wide range. In either kind of difficulty we
may be able to assign an upper bound for the numbers concerned, but if we scale them so that the upper
bound is brought within range we may lose a great deal of accuracy, especially if an upper bound can be
assigned only roughly. It is worth pointing out that we can often choose what seems to be a reasonable
upper bound for each quantity and scale it accordingly. If the bound is ever exceeded overflow will
occur, and this can be used to prevent wrong results passing undetected. The overflow-indicator is a
powerful aid to correct scaling. The scaling difficulty is reduced in computers having a long word; in
Pegasus we have 38 binary digits after the point (corresponding to rather more than 11 decimal digits)
and we can often afford to scale down fairly generously without losing too much accuracy. If necessary
we can work double-length and represent all our numbers by two words, giving the equivalent of nearly 23
decimal digits. The alternative approach of floating-point^ working is usually to be preferred for
general use, since it usually solves all scaling problems; in fact we need not scale the numbers at
all. In contrast, the ordinary mode of operation may be called fixed-point.

It is common in scientific work, and in engineering, to represent very large or very small numbers
by introducing a power of 10 as a scaling factor. For example, we may write numbers in the form

-3.457 X lo"-^ or 8.664 x 10"^.

Here we arrange to write the decimal point immediately after the first significant digit of the number
and we introduce a suitable power of 10 to correct the value. We can use a similar technique inside the
computer; but since Pegasus is a binary machine it is usually easier to, associate a power of 2 with the
number, instead of a power of 10. A quantity x is consequently represented in the form

X - .1.2"

where a is an integer chosen so as to bring A within some convenient range. We shall call A the
argument of the number x, and a the exponenftt. Each quantity x is represented inside the computer by
its argument and its exponent, i.e. we store two numbers to represent one quantity; there Is no need
to store the base 2 of the quantity because this will be the same for all. We shall restrict the
argument A to one of the following ranges:
(a) Vi < A < ^ it A > 0,

or (b) -%< A < -»/4 if A < 0.

f The binary point is allowed to float, its position being automatically adjusted to suit the size of
the number. In French this is virgule flottante, in German gleitendes Komna.

tt There is unfortunately no generally accepted terminology for these two parts of a floating-point

number. The quantity A is sometimes called the aianiissa or the numerical part; and a is sometimes

called the index.

192

SOME PROGRAMMING TECHNIQUES 9.1

The number zero requires special treatment in all floating-point work. The advantage of choosing these
ranges is that two such arguments can be added or subtracted without overflow. The exponent a is a
suitably chosen signed integer.

A sequence of orders will be needed to do an ordinary arithmetical operation on two floating-point
numbers. Suppose, for example, we are given two such numbers

X = ^.2" and y = B.2^,

and we wish to form their product z - C.2^ . We have

z - x.y = {A. 2") (B.2^) = >1B. 2""*■^

so that we can put C = AB and c = a+6. This is quite simple to work out, but the operation is not yet
complete. The quantities C and c must now be adjusted so that, provided C is non-zero, either

(a) 1/4 < C < ^ if C > 0.

or (b) -1/2 < C < -% if C < 0.

If this is not done the argument of the result will be less than the argument of either of the original
operands; and after a sequence of operations severe scaling down may occur. In other operations the
argument of the result may be larger than that of either of the operands. The argument of the result
must, in general, be shifted up or down to bring it into one of the above standard ranges, and the
exponent must be simultaneously adjusted by subtracting from it the number of upward shifts. This
opeiration is called normalizing the floating-point number and, after it has been carried out, the number
is then said to be in normal form J The following are two examples of floating-point numbers which are
not in standard form:

% X 2^ and " n x 2^°,

The result of normalizing these numbers is, respectively,

% X 2^ and -% x 2'^.

The normalizing process can, of course, be programmed by using only the orders so far described.
This is, however, very inconvenient and, what is much worse, relatively slow; a special order, with
function 56, has therefore been provided. 'This order does the whole process automatically; it deals
with a double-length argument. We put the argument of the floating-point number into the double-length
accumulator, X6 and 7, and the exponent into one of the other accumulators, say Xi. When the order

® 4 56

is obeyed the floating-point number is automatically normalized. The iV~address in the order shows the
maximum number of places through which the argument is to be shifted; it is in fact shifted up at most
N-1 places. This limit has been introduced primarily to^ allow for the argument being zero. The 56-order
may be described verbally as follows.

56 (normali ze) Normalize the floating-point number whose argument is the fraction (pq) in X6 and 7,

and whose exponent is the integer in the specified accumulator. But do not shift
the argument more than N-1 places up (N being the number written first in the order).

The 56-order may shift the argument down one place (and add one to the exponent), or it may leave the
argument and exponent unaltered, or it may shift the argument up a variable number of places and subtract
this number from the exponent. Usually the possibility of a zero argument must be specially allowed for
(by including a 60-order to a special sequence of orders, for example), and it is usually ruled out by
the time the 56-order is encountered. Consequently it is normally enough to write a large number (such
as 80) in the ^V-address of the order.

The effects of the order may be defined algebraically as follows (here X is not 6 or 7):

(pq)' - 2^ (pq). (q^ 0).

x' = X - fi (integer convention),

where /x (Greek letter mu) is an integer chosen so that

- (a) % ^ (pq) ' < 1/2 and -1 ^ /x ^ N-1.

or (b) -Yz < (pq) ' < -Vi and -1 ^ /u, < iV-1,

or (c) -Vi < (pq)' < 1/4 and Ai = N-1.

Only one of these three statements can hold; the integer /U. is uniquely determined. In these
inequalities (pq) is, of course, a double-length fraction. The shifting which occurs in the double-
length accumulator is similar in its effects to that produced by a suitable 54- or 55-order (the

' The terms s tandardizing and in standard form are also used.

193 -

9-1 SOME PROGRAMMING TECHNIQUES

sign-bit of 7 is always cleared by a 56-order, however) +. A 56-order can set the overflow- indicator,
but only if the exponent overflows (this is very rare).

As an illustration of the use of a 56-order, we give a sequence of orders for multiplying two
floating-point numbers whose arguments are in X6 and XI. respectively, and whose exponents are in 14
and X5.

6 7 20 multiply arguments

5 4 01 add exponents

r s) 4 56 normalize

We assume here that the two operands are in normal form and are not zero. There is then no need to put
N greater than 3 in the 56-order. The above sequence leaves the exponent in J4 and the argument double-
length In X6 and 7; to put into X6 the best single-length value we must round it, which can be done by
adding the following two extra orders (see Sec. 3. 4):

add %e

But if we do this we should normalize the result again.

The division of two floating-point numbers is done similarly, we have now to divide the arguments
(first doubling that of the divisor to prevent overflow) and to subtract the exponents. In fact

(/1.2«)/(B.2^) = (A/B).2''-^ = (/l/2B),2°-^ + ^

The following orders can be used, given the two operands as above:

loop stop if divisor is zero

double argument of divisor

divide arguments

(pq) is rounded quotient A/2B

7 60

7 01

6 26

6 00

7 00

4 03

4 41

4 56

a - b + 1

normalize quotient

We assume that the operands are in normal form and that the dividend is not zero. There is no need to
put N greater than 2 in the normalize order. This sequence leaves the argument (single-length) ++ in
X6 and the exponent in 14 in normal form.

The addition or subtraction of two floating-point numbers is a little more complicated than
multiplication or division. Suppose we have to add

X = A. 2" and y = B.2^.

We first of all compare a and b, the two exponents. If a ^ 6 we proceed, but if a < 6 we interchange
the arguments and exponents first. Assuming therefore that a ^ b yte must shift B down through a-b
places (thus effectively aligning the binary points) and then add it to A. We can write

C.2'' = /1. 2° + B.2^.

= (A + S.2^*°)-.2°.

We can therefore put the exponent c equal to a, the larger of the two original exponents. The process
for subtraction is similar but we must change the sign of B first. The following sequence can be used
to add or subtract two floating-point numbers (as above, the exponents are in 14 and 15 and the arguments
are in 16 and 17). In this sequence d denotes a-b or b-a, whichever is not negative; there is a trick
to prevent a shift of more than 38 places. This sequence shows a good application of a modified shift
order, perhaps the most important one.

V t The sequence of events when a 56-order is obeyed is roughly as follows: the exponent enters the
mill and 2 is added to it; the double- length argument is shifted up (if necessary) until its two
most-significant bits are different, it is then halved; the exponent is reduced by 1 for every shift
(including the last) and is finally sent back to its accumulator; the value of N is tested and if
non-zero, reduced by 1 for each doubling of the argument; if N is ever zero the order is immediately
terminated by halving (pq) and subtracting 1 from the exponent. Overflow of the exponent is sensed
only at the end of the order, when the exponent is sent back to its accumulator. The net result if
;v is is that (pq) is always halved and one is added to the exponent. If A^ is 7 then p' is the Ipff
half of the normalised (pq) and q' = q./j.. if ^ is 6 then p' = p./j. and q' is the right half of the
normalized (pq). The 56-order can be modified in the same way as the other orders of group 5 but
this is only rarely useful. The time taken is similar to that of a shift order plus two word-times
/k i.e. a total of /j. + 5 word times if it is an a-order, or ,a + 4 if it is a 6-order.

" Strictly speaking we should round the argument as was done after a multiplication since the normalize
order may have shifted the quotient down one place, thus putting one binary digit into XT.

- 194

SOME

PROGRAMMING

TECHNIQUES

9.1

subtract

-(►

0,0

add

7
4
0.4
5
5
7
6
7

0.5+

@
®

7
5
5
4
5
6
7
6
5
5
5
7
6
7
4

--
62

"oT

00
56

Negate B

a- b to A'5

jump if a ^ fc (here d - a-b ^ 0)

b to J4

b-a >

interchange
A and B

interchange
operands if
b > a

d-38

jump if d ^ 38 (with in 5m)

d-38 to 5m if rf < 38

shift down d places (but at most 38 places)

add arguments

normalize

We again assume that the result is not zero. This may occur however because of exact cancellation, and
we may get nearly complete cancellation of the two arguments leaving a result much smaller than either
of the operands (this can happen with normalized operands only if the two exponents are equal or differ
by 1), In this way serious loss of accuracy can arise.

The sequences of orders given above are meant to illustrate some of the ideas which can be applied
to floating-point work. They are not complete because the number zero has not been provided for. In
these sequences we have assumed that the arguments and exponents of the operands are available in certain
accumulators. In practice it is usually wasteful to allocate two words to each floating-point number.
This is because the exponent is seldom very large in absolute value. It would be unusual, for example,
to have a number as big as 2'-°° (which corresponds to about 10^^°) or as small as 2"-^*^'^. We can therefore
represent the exponent quite adequately with only 8 or 9 binary digits. Again, in most applications
the argument need not be given a whole 39-bit word; this is because it is normalized, which means that
nearly all its digits are significant digits - we do not get strings of zeros or ones before the first
significant digit as we do in fixed-point work. It is therefore possible for the argument and exponent
to be packed into a single word; usually the argument occupies the first 30 binary digits, and the
remaining 9 represent the exponent. With 9 bits to represent the signed exponent, a, this can be
allowed to assume values in the range

-256

a < 255.

This corresponds to allowing numbers to get as large as about 10^^. (This is about the number proposed
by Eddington as the total number of protons in the universe!) The argument is represented with a
precision corresponding to a little more than 8 significant decimal digits.

A standard packed floating-point number is represented by a single word in which digits to 29
represent -4, the argument of the number, directly: and digits 30 to 38 represent a + 256, thus:-

30 bits

9 bits

A a + 256

There are several advantages to be gained by packing a + 256 into the word, rather than a; the main
point is that a + 256 is never negative. Such a number will of course, have to be split up. or unpacked,
into its component parts before we can do any arithmetic with it. The result of an arithmetical
operation will also have to be packed up into this form before being stored. These packing and unpacking
operations are simplified if the exponent is held as a + 256, and the extra complication in the
arithmetical sequences is slight (in fact the addition and subtraction sequence given above need not be
altered at all).

There are a number of Library subroutines and programmes which handle these packed floating-point
numbers; there are subroutines for input and output, for arithmetical operations, elementary functions
and certain processes. With many of these the number of binary digits, n, used to represent the
exponent is determined by a preset-parameter and can be given any reasonable value (with all these
routines there is an optional parameter-list to set n equal to 9. the most generally used value). If n,.
bits are used, they represent

a + 2"-l

195

9.1 SOME PROGRAMMING TECHNIQUES

where a is the exponent which can take values in the range

-2""1 < a ^ 2"-l - 1.

The number zero is represented in all these routines by an argument of zero and an exponent which is as
small as possible. Thus if 9 bits are used for the exponent (n = 9), zero is represented in the form

0.2

-256

which is called the standard zero. The word representing this has in the part allotted to the
argument; the exponent digits represent 256 - 256 = and are also zero; the word therefore has in
every digital position and is a null word. In fact the standard floating-point zero is represented by
the same word as a fixed-point zero. We can still use XO to supply zero even if we are doing floating-
point, work. The first four Jump orders (60 to 63) also have their usual effects, even if the number
being tested is a packed floating-point number.

If, during the course of a calculation, the exponent of some quantity becomes larger than 255 (or,
more generally, larger than 2""^ - 1 if n bits are used for the exponent) then floating-point overflow
is said to have occurred. This is an unlikely event and is usually due to a mistake in the programme.
If an exponent less than -256 (or. in general, -2""^) is obtained this is called underflow; it is then
usual to set the result equal to a standard zero. If there is exact cancellation of the arguments the
result is also made into a standard zero.

We give below an example of a complete floating-point programme. This is designed to read in a
list of k numbers x^ followed by a single number y from an Input data tape and to evaluate and print
the function

= AyO

H)} = v/{3

2 X.}.

i-l I

The value of k is determined by counting the numbers x^^ read from the tape; after the last of them
(xf^) an L is punched. This is followed by the single number y. These numbers are all punched in their
usual form with the decimal point where required among the digits; they are read in by Library sub-
routine R 101. The floating-point additions and the multiplication are done by R 610 and the square-
root is found by R 611. The answer Is finally printed by R 11 and there is then a loop stop.

When R 101 is used first in this programme it reads in numbers until the L on the data tape is
found. Each of these numbers is stored in the location whose address is 5j)^; this is increased by
unity after each number has been stored. Note the computing store links and partial cues used with R 610.

D
N

FUNCTION Z
Al

-0

101

-0

101

R 1

-0

610

R 1

-0

610

R 1

-0

610

R 2

-0

611

R 2

-0

>■ Calls for cues.

196 -

SOME

PROGRAMMING TECHNIQUES

9.1

E 2.0

►

5.7

0.0

0.2

0.3

5.7

0,7

1 +

0.0

Pinal T.A. to "byi
set link

+ cue 02
to R 101

► link

Read in
k numbers

to main
store.

collating mask i^'^o^^

> ^c

5.1j^ = address of first number, 5.7^ = k

+ cue 01
to R 101

link

Read in

y
to X6

0.0

{Ho

'- »»

. ^ P-

fo.o

■■""
6

fc!.

2.0

-Q-

3.0

0.6

0.3+

0.0

>- to 0,0 (temporarily)
set 2^1 and 2c

R 610 to f/3,4,5
(+ partial cues)

zero to XG

first numbers to f/2

xi to Xl

set link

Add X;

count down from k

y to XI

set link for R 610

Add all

using
R 610

197

9.2

SOME

PROGRAMMING TECHNIQUES

3.0+

1 1.0

1. t,

1.2
+0

1 2

0.3

0.6

®.

1 ^

' kC*

10.6+

multiply Zxi by y

^ + cue 01
to R 611

/{ySx-} to X6

result to XI
set link in X6

programrae-
parameters

+ cue 03
to R 11

link (gives loop stop)

Print
result

A2
A3
E 2.0

Note that in this programme the main set of numbers x^ is placed in the main store in consecutive
locations starting at the final value of the Transfer Address at the end of the input stage of the
programme. We thus avoid fixing any main store addresses; indeed the only absolute address in the
whole programme is that in the final E-directive. This means, however, that we cannot safely restart
the programme by a simple E-directive; but we can easily use instead a pair of directives such as the
following, if the need should arise:

T 100.0

E 2.0.

9. 2 Interpretive and conversion programmes

Floating-point work is an example of what is often called programmed arithmetic. We have to obey
quite a sequence of orders, in fact a subroutine, to do even an addition or a subtraction. This is
because the numbers on which we are operating are not the simple numbers round which the computer was
built. t There are other examples of awkward kinds of quantity, complex numbers, or double-length or
multi-length numbers for example, where the position is similar. By the use of carefully written
subroutines most of the serious difficulties can be removed, and the main programme becomes almost
entirely made up of organisational (or red tape) orders setting links and parameters, calling in the
subroutines in the appropriate way, effecting the transfers of programme and numbers, counting,
modifying and so on. The actual programme is in fact quite complicated, even if the original flow-
diagram was simple. The maze of organisational orders may conceal the essential simplicity of
structure of the prograinrae; so much, in fact, that it may become quite difficult to prepare, test
and correct it. Yet an attempt to write the programme without the use of proper subroutines would
introduce other, more intractable, problems and make the programme even more lengthy.

A possible approach is to sketch out the programme first in some suitably chosen abbreviated
notation; this can later serve as a guide while the actual programme is written. For example, in
floating-point work we could number all the locations used to hold floating-point numbers (assuming
these are all held in the main store); the contents of these locations could be referred to as variables
and written vq, v^^, vg, v^,.. We could then indicate the floating-point addition of say v^ and v ^2 by
writing down

which means that the 8th variable is to be set equal to the sum of the 5th and the 12th. A whole
programme may be written down in this sort of way, one "order" below another. Later on we can go
through it and translate each of the "orders" in it into the proper computer orders to extract the
variables, set any parameters and links, call in the subroutine appropriate to the operation in the
"order" and dispose of the result. If we suitably choose the way in which the original "orders" were
written down, this translation process can be made systematic and subject to definite rules; in fact
it can be carried out by someone who is not familiar with the technique of programming, provided only

t Some machines, e.g. the Perranti Mercury Computer, have built-in floating-point operations; on such
machines a single order can be used for an arithmetical operation.

- 198

SOME PROGRAMMING TECHNIQUES 9.2

that the rules to be followed are precise, clearly expressed and cover all cases. Indeed we can get
the computer itself to effect the translation, if we supply a suitable programme incorporating all the
rules.

The general procedure is to write our programme in a convenient language well adapted to the
nature of the work, and to punch it on tape much as it is written. This is the pseudo- pro gramme; the
tape is then read by the computer under the control of a special conversion programme and converted
into a programme of machine orders^. The result of the conversion is an ordinary programme and the
computer can either obey it immediately or punch it out (probably In binary) for later use. The
Initial Orders are in effect a rudimentary conversion programme, because we write and punch our orders
and numbers in a fairly convenient way, and these are converted into the binary form required by the
ccffliputer. There is in fact quite a lot of conversion done by the Initial Orders (consider the effects
of reading A3, for example), but a full conversion programme does a great deal more.

There is another way of handling the problem of translating a programme from a convenient
language for a particular problem into the ordinary machine language. As before, the programme is
punched as written and read in, but this time the "orders" on the tape are not converted into machine
orders; instead they are stored with only the minimum of conversion. For example the "order" given
above could be stored in the form of three addresses specifying the variables together with a single
integer specifying the function - all these could be packed into one or two words. When the end of
the tape is reached the whole programme will have been stored, but only in a simple coded form corres-
ponding closely to the written pseudo-programme. At this point a special programme is called in,
which can extract each order from the store and interpret it, i.e. carry out the operation called for.
For example, the addresses in the order could be used as modifiers to select the operands and to write
away the result; the function number could be used to modify a special cue which would then bring in
the appropriate subroutine. In this interpretive scheme the "orders" are each carried out as soon as
they have been translated, and the programme then passes on to interpret the next "order". Thus if
there is a loop in the programme being interpreted, each "order" in the loop will be translated each
time the loop is traversed; whereas in a conversion scheme the translation process is done once and
for all before any of the "orders" are obeyed. An interpretive scheme is consequently slower, in
general, then a conversion scheme. On the other hand each "order" for an interpretive scheme occupies
only a word or two in the store whereas the converted programme of machine orders resulting from the
use of a conversion scheme will occupy a considerable amount of space. The extra time spent inter-
preting orders in an interpretive scheme is the penalty paid for compactness of the programme; there
are circumstances when this time does not matter, for example when the individual operations carried
out take much longer than the interpretation time.

In either type of scheme a convenient external language is ultimately translated into the internal
machine language. There are other ways of doing this. All such schemes are often described collectively
as automatic programming, though perhaps automatic coding is a term to be preferred since the calcula-
tion usually has to be carefully planned. Some automatic programming schemes can be considered as
directed at removing some difficulty or disadvantage in a particular computer, but it is not easy to
make any firm definitions. Some of these schemes are so successful that they are regarded as the normal
way of putting a programme into the computer; usually, however, the machine-made result is inferior to
that produced by hand. An experienced programmer can make use of many special features of a problem to
make his programme efficient; it is very difficult to cater for such individual features in an
automatic programming scheme designed as a rule to handle a wide variety of problems. In general,
therefore, the use of an automatic programming scheme can be expected to slow down the computer - and
by a factor which may be as large as 100 in unfavourable cases. The programming time for a particular
problem may well, however, be reduced in about the same ratio and the programme will often be correct
at the first attempt. As an important consequence, the overall time from beginning to study a problem
to the obtaining of accurate results is usually dramatically reduced, perhaps from months to hours.
This is especially true of what might be called small problems, which can be fairly simply stated and
do not involve large masses of data or results.

The kind of external language chosen for one of these schemes depends very much on the operations
to be carried out, the nature of the operands, the frequency with which the scheme is to be used and
the programming experience of the users. It is, of course, restricted to the characters which can be
punched in the paper tape and printed. A number of schemes are available for Pegasus and some of
these are described below. If a great deal of a certain kind of work is to be undertaken then it is
worth considering the writing of some special automatic programming scheme. Such a scheme, if written
with some special aim in view, need not be a very elaborate one; and some technique might well be
found peculiarly well adapted to the subject. For example, what is known as a tape-steered programme
finds some applications: this kind of programme is designed to read in an "order", or some parameters,
or perhaps Just a J-directive from the input tape, and to carry out some operation (on internally
stored information) the nature or details of which depend on the information read in. When the
operation is complete another "order" or other information is read in. A scheme like this is
reasonably flexible and has the advantage that the pseudo-programme does not occupy any storage space
in the computer; though of course loops are impossible.

The user of an automatic programming scheme will usually never know into what machine orders his
programme is ultimately translated; he is, as it were, removed or insulated from the actual machine
and thinks only in terms of the external language of the scheme. In fact to him the computer appears
as quite a different machine, perhaps capable of doing the most complex operations when it obeys a
single order; he need not know anything of ordinary prograiwning at alltt. Ttiere is, however a
difficulty which can arise if the user' s programme and that actually obeyed by the computer are
widely different: if the programme does not work then it may be quite difficult to find out why.
However, the programme is much more likely to be free from error than an ordinary computer programme,

' A machine order is simply an ordinary order as obeyed by the computer.

tt An amusing distinction has been drawn between the Primitives of programming, who think always in
binary or octal and like to keep as close as possible to the computer way of doing things, and the
Space Cadets, who like to punch algebraic equations, English words and integral signs on their tapes.

199 -

9.3

SOME PROGRAMMING TECHNIQUES

and most schemes provide facilities for extra informative printing at the start which can later be
omitted.

9.3 The Autocode

The Pegasus Autocode is a conversion scheme which greatly facilitates the programming of certain
kinds of problemt. It Is especially suitable for such technical and scientific calculations as the
evaluation of formulae, the tabulation of special functions and small ad hoc calculations; but it can
also be used in commercial and industrial work for the testing out of complex flow-diagrams, and has
many other applications. The user of the Autocode need not know anything of ordinary Pegasus programming,
though there are certain facilities which he cannot otherwise use.

To use the Autocode on a particular problem the calculation must first be broken down into a
sequence of steps, each of which is the calculation of a number from one or two previously calculated
numbers. Each step is written as an order or instruction and the whole sequence forms the programme of
the calculation. The programme is punched, as written, on a teleprinter or a keyboard perforator and
the resulting tape is then read into the computer by the Autocode scheme (Instruction Input section),
which converts the programme into a suitable internal form, which is stored. At the end of the tape'
are some special symbols which cause the programme to be obeyed. At various stages while it is being
obeyed the programme can cause printing of results (they are actually punched, as usual, and printed
later), or it can read in numerical data from tapes in either of the tape-readers.

There are two kinds of numbers which can be handled by Autocode Instructions:

(a) Variables, denoted by vO, vl, v2, v3, ... These are actually standard, packed, floating-point
numbers with 9 bits for the exponent, and may therefore lie in the range

-225't < „ < 2"'^ (1 - 2-29).

They are represented with a precision corresponding to between 8 and 9 significant decimal
digits, except that numbers in the range

-2-257 < „ < 2-257 (1 . 2-28)

are treated as zero,
(b) Indices, denoted by nO, nl, n2.

These are signed integers in the range
-8191 < n < 8191.

They are actually stored in the modifier -posit ion of words and are handled in fixed-point

form.
Most of the numbers entering into a calculation are variables; the indices are introduced as auxiliary
quantities, mostly to facilitate counting and modifying. The v and n symbols are available in figure-
shift on the teleprinter; the digits which follow them are to be thought of as suffixes although they
cannot be printed as such on the teleprinter. The total available number of variables and indices is
adjustable (by preset -parameters, in fact), but they are normally fixed so that they are

(a) 1380 variables written vO, vl, ... , j;1379, and

(b) 28 indices written nO, nl, . . . , n27.

Most of the Autocode instructions take the form of an equaticM giving the new value of a variable
(or index) in terms of one or two numbers or previously calculated variables (or indices). For example,
the instruction

vl = i;2 + w3

means that the new value of vl is to be the sum of v2 and vS. The Instruction

v5 = v5 + vl

means that the new value of v5 is to be the sura of vl and the old value of v5.
instead, of variables on the right of the equality sign; thus the instruction

Numbers may be put

vB = i;2 + 35.771

means that the new value of v8 is the sura of v2 and 35.771. As an example of a sequence of these
instructions dealing with variables let us evaluate

31.41t;5 - v922 +{(-6.535»;8 + 1,97 - 9)/v323}.

and put the result in vO. The following sequence of instructions can be used (here we use vl as working
space) :

31.41 X v5

v92 X i;92

vO - vl

-6.535 X v8

vl + v97

vl - 9

vl/v323

vO + vl

t The scheme is based on that of Dr. R. A. Brooker for the Ferranti Mark I computer at Manchester University
although it differs greatly in detail,

Brooker, R. A. An attempt to siaplify coding for the Manchester Electronic Computer. Brit. J Ann
Phys. (1955), 6, P.307.

- 200 -

SOME

PROGRAMMING TECHNIQUES

9.3

Note how each instruction involves only two variables or numbers on the right. We can also use
instructions like the three following ones, which have only one variable or number on the right:

v8 ^ v4, v6 = -v2, vB = 43.

There are exactly analogous instructions for handling indices, for example:

n3 = n8 + n3, n2 = -79 - n5,

nO = 288, n4 = -1971,

but it should be remembered that indices can take only integral values. In general we cannot mix indices
and variables in the same instruction; a few sim.ple instructions like this have, however, been provided.

The purely arithmetical instructions in the Autocode are summarized in Table 9.1. In this table
vl, v2 and v3 are given simply as representative variables, and nl, n2 and nS represent any three
indices. The instruction nl = n2/n3 gives the integer quotient when n2 is divided by n3, and the
instruction nl = n2*n3 gives the corresponding remainder. (The analogous instructions with a minus
sign give the same numbers with their signs changed). In any of the instructions any variable (or index)
on the right-hand side of the equality sign may be replaced by a positive number (or integer).

Certain elementary functions may be evaluated by a single instruction; for example we can write

vfy = SQRT t;9

to evaluate a square root, or

"SIN v99

to evaluate a sine. The available functions are tabulated in Table 9.2; they all apply to variables
only, except for MOD which can be used to form the modulus of an index or a variable. Again, any
variable written on the right may be replaced by a positive number; for example:

v37

LOG 5. 1058209.

Variables

Indices

vl. = v2 vl =

-v2

n2 nl = -n2

vl = v2 + v3 vl =

-v2 + v3

n2 + n3 nl = -n2 + n3

vl = v2 - v3 vl =

-v2 - v3

n2 - n3 nl - -n2 - n3

vl = v2 X v3 vl =

-v2 X v3

n2 X n3 nl = -n2 x n3

vl = v2/v3 vl ^

- v2/v3

n2/n3 (quot) nl = -n2/n3

n2*n3 (rem) nl = -n2*n3

Mixed: nl -

-v2 (nearest integer)

vl =

-n2

vl =

n2/n3

-n2/n3

Table 9. 1.

Autocode instructions - arithmetic.

vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
nl

MOD v2
INT v2
PR AC v2
SQRT v2
SIN v2
COS v2
TAN v2
CSC v2
SEC v2
COT v2
ARCSIN v2
ARCCOS v2
ARCTAN v2
LOG v2
EXP v2
EXPM v2
MOD n2

vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
vl
nl

-MOD v2
-INT v2
-PRAC v2
-SQRT v2
-SIN v2
-COS v2
-TAN v2
-CSC v2
-SEC v2
-COT v2
-ARCSIN v2
-ARCCOS v2
-ARCTAN v2
-LOG v2
-EXP v2
-EXPM v2
-MOD n2

modulus
integral part
fractional part
square root

circular functions

inverse

circular functions

natural logarithm
exponential
exp (-v2)
modulus of index

Table 9.2 Autocode instructions - functions.

201

9.3

SOME PROGRAMMING TECHNIQUES

As usual. Autocode instructions are obeyed in the order in which they are written, until a jump
instruction is encountered. If a jump occurs, the next instruction to be obeyed is identified by its
label, which is simply a small positive integer written in front of the instruction and separated from
it by a right bracket. For example,

7) v4 = -9. 44/ 1/5

is labelled 7. Any instruction can be labelled, but the same label should not be used twice,
required we can attach two or more labels to the same instruction, thus

9)2) v3 = LOG vO

The first instruction in an Autocode programme is always automatically labelled (there is no need to
write this label in). A jump instruction always includes an arrow; the simplest is the unconditional
jump, for example

-7

is the Autocode instruction meaning; jump to the instruction labelled 7.
conditional, however; consider the following instruction

- 8, vl ^ vG

Most of the jumps are

This means: jump to the instruction labelled 8 if vl ^ vG, otherwise carry on with the next instruction
as usual. The following three instructions resemble this one closely:

- 8, vl 5: ~v6

- 8, -vl > vG

- 8, -vl > -v6

These four instructions can be summarized as

+ vl > ± v6

All the available jump instructions are summarized in this way in Table 9.3; as usual the numbers 1,
2, 3 merely stand for any convenient numbers. There is a great variety of jumps, and the scheme is
consequently very flexible. In a jump instruction, any variable may be replaced by a number, or any
index by an integer. The following are examples of this:

- 4. ^ v62

- 8. n6 ^ -20

The jumps including tests of approximate equality are intended to allow for the effects of the rounding
errors inevitable in most floating-point work; care should be taken to set the valiie of nO before
using them (e.g. by an instruction such as nO = 20).

- 1

(unconditional

jump)

- 1.

±v2 > ± v3

- 1, ± v2 > ± v3

-» 1,

± v2 = ± v3

- 1, ± v2 7^ ± v3

- 1.

± n2 ^ ± nS

- 1, ± n2 > ± n3

-* 1.

± n2 = ± n3

- 1, ± n2 ^ ± n3

- 1,

± v2 = * + v:

t (jump

if approximately equal; more exactly.

jump

if the two variables agree to nO significant

binary digits - i.e. to about 0.3 x nO signifi-

cant

decimal figures)

-* 1,

+ v2 ?^ * ± v3

(jump

if not approximately equal)

Table 9.3 Autocode instructions - jumps.

As an example, let us suppose that we are given two positive numbers vl and v2 and we have to
replace them, respectively, by

y2(vl + v2) and /(vl x v2) .

This process is to be repeated until the two quantities are nearly equal, say until they agree in their
first 20 binary places. The result is in fact an approximation to the Gauss arithmetico-geometric
mean of the two quantities. The following sequence of orders can be used.

- 202

SOME PROGRAMMING TECHNIQUES 9.3

nO = 20

3)t;3 = vl

vl = vl + i;2

vl = 0.5 X vl

i;3 = v3 X v2
v2 = SQRT v3
-3. vl ^ * v2

The first instruction sets nO for subsequent "approximately equal" tests. The variable v3 is used to
hold intermediate quantities, it is first used to keep the value of vl.

The technique of modification can be applied to any Autocode instructions involving variables.
For example, we can write the instruction

v8 = ^998 + vn6

which means that the new value of the 8th variable is the sum of the 998th variable and the n6-th
variable. The n6 in the variable vn6 is to be thought of as a suffix (i;„ ) though it cannot be typed
in this way on a teleprinter. Thus the Instruction written above is equivalent to the instruction

v8 = t;998 + v28
if nG = 28 at the time the instruction is obe3fed. Any variable can have this kind of suffix, for example

7;wO = vn3 + vn4:

We can also write more complicated kinds of suffix, for example

v(8 + n2) = v(53 + n4) + v(-2 + nil)

in which each bracketed expression is regarded as a suffix. The integer (h, say) in this kind of suffix
must be written before the n and can have any value in the range

-2048 ^ /i ^ 2047,

but the result after the addition of the index n must not be negative. The index must be added, never
subtracted.

We can also have modified jump Instructions, of which the following are specimens:

-* nl, vO > v61

- n9, -n8 > -21

- (-4 + nS), nO 7^ 8

Instructions like these are not often needed.

A small group of instructions is provided for the inpnt of numbers. A simple one is the following:

v6 = TAPE

This causes v6 to be set equal to a number read from the input tape in the main tape-reader. 'Oie
instruction

v5 = TAPE 13

causes 13 numbers to be read in and placed in v5, v6, .... vYl. Tlie instruction

t;28 = TAPE *

causes numbers to be read in and placed in v28, v29, v30, ... until an L-directive is read. There are
a number of other input instructions; they are all included in Table 9.4. It should be noted that
any input instruction has the following properties:

(a) input ceases and the next instruction is obeyed when L is read,

(b) nO is always put equal to the number of numbers read in,

cannot be used with indices).
Suppose as an example, that the instruction

^■1 = TAPE ♦

- 203

9.3 SOME PROGRAMMING TECHNIQUES

is obeyed and that the following tape is in the main tape-reader:

DATA 3

+1.5

Q +10 -0.657 +0.9876

Q -15

+0.55321

- 13

When the instruction has been obeyed the following will be stored:

vl = +1.5, v2 = -0.657 X 10^°. v3 = +0.9876 x 10^°,

t;4 = +0.55321 x 10"^^. v5 = -13 x 10" ^5, nO = 5.

The name DATA 3 will be printed when the tape is read. Each number on the tape is punched as written,
immediately preceded by its sign, and must be terminated by Sp or CR LP (if indices are to be read in,
by ao instruction like n3 = TAPE, for example, they must be punched without a decimal point).

TAPE

read in one number and set in vl

TAPEB

TAPE
TAPEB

read in one integer and set in nl

vl
vl

TAPE n2
TAPEB n2

;

read in n2 numbers and set in vl, v2, ...

TAPE nl
TAPEB nl

}

read in nl integers and set in n2, n3, ...

vl
vl

TAPE *
TAPEB *

•

■

read in numbers up to L on tape; set in vl, v2, ...

nl
nl

TAPE *
TAPEB *

■

read in integers up to L on tape,
set in nl, n2, . . .

TAPE Instruct

ions

'fer to the main tape-reader.

TAPEB to the

second

tape- reader.

Table 9.4 Autocode instructions - input

There are two main methods of output in the Autocode. The most used consists of a special
instruction, of which the following is an example,

PRINT v6, 3045

This instruction causes the value of v6 to be printed in a way determined by the style-number 3045. To
find the style-number to write in a print instruction we must first decide whether the number is to be
printed on a new line or on the same line as the previous number, and whether it is to appear in
floating-point form (i.e. as a number and a decimal exponent) or in fixed-point form (with the decimal
point in its proper position). This determines the first digit (a) of the style-number, which can be
found from the following table:

ating-point

Fixed-point

a = 1

a = 3

a = 2

a = 4

Print on a new line (CR LP (p before number)

Print on the same line (Sp before number)

The rest of the style-number is determined by the number of digits (6) to be printed before the decimal
point and after it (c); the style-number is in fact

1000a + 206 + c.

so that style 3045 causes the number to be printed in fixed-point (or normal) form on a new line with
two digits before the decimal point and five after it. The number is preceded by its sign and is, of
course, properly rounded. If desired we can write the instruction in the form

PRINT v3, n9

for example, when the current value of n9 is taken as the style-number. We can also use a similar

- 204

SOME

PROGRAMMING TECHNIQUES

9.3

instruction for printing indices; here only the first digit of the style-number matters and the index
is always printed as a 4-digit signed integer. Thus the instruction

PRINT n3, 4000

causes the value of nS to be printed, preceded by a single space. In all printing, left-hand zeros in
the integral part are, of course, suppressed (i.e. replaced by spaces).

An alternative method of output is mainly useful to provide extra printing in the development stage
of an Autocode programme. We can write XP (for printing on a new line) or SP (for the same line) before
an instruction; this will cause the result of the instruction to be printed. Thus the instruction

XP i;9

t;3 + v5

causes the new value of v9 to be printed on a new line after the rest of the instruction has been
obeyed. The printing caused by XP and SP is optional; it can be suppressed by depressing the sign-
digit key of the handswitches. We can alternatively put X or S in front of an instruction; these cause
printing (non-optional) of CR LP or a single Sp, respectively, and can be used to lay oUt the printing.
We may put XP, SP. X or S in front of any instruction which carries out an arithmetical operation or
evaluates a function.

The output instructions are given in detail in Table 9.5.

PRINT i;l, n2 Print vl in style number n2 = 1000a + 206 + c.

If a = 1 print CR LP then number in floating-point form with b digits before
point in argument and c digits after point, then Sp and two-digit
exponent, then Sp Sp. Width of printing 8 + 6 + c.

If a = 2
If a = 3

If a = 4

print Sp then number as for a = 1. Width 9 + 6 + c.

print CR LF <p then number in fixed-point form with 6 digits before
point and c digits after. Width 2 + b + c. But if 6 is too small
print in floating-point form as if a = 1.

print Sp then number as for d = 3. Width 3 + 6 + c.
small print as if a = 2.

If b is too

PRINT nl, n2 Print nl in style number n2 = 1000a

If a = 3 print CR LP then 4-digit index. Width of printing 5.

If a = 4 print Sp then 4-digit index. Width = 6.

XP before an instruction (arithmetical or functional) the result of which
is a variable:

Obey instruction, then print CR LP and result in floating-point form;
the argument to 9 decimals with its first significant digit after the
point, then Sp and two-digit exponent, then Sp Sp. Width of printing
18. There is no printing if HO = 1.

XP before an instruction the result of which is an index:

Obey instruction, then print CR LP and result as four-digit integer.
Width 5. There is no printing if HO - 1.

SP is similar to XP but Sp is printed instead of CR LP0. Width 19 for
variables, or 6 for indices.

before any arithmetical or functional instruction:
Sp respectively, after obeying the instruction.

Print CR LP or

Table 9.5 Autocode inntructions - output.

The following are typical results of some instructions causing output.

PRINT vnlO, 1064
PRINT t;3, 2064
PRINT vi. 2044
PRINT v9, 3062
PRINT i;(l+n2), 4026
PRINT n2. 3000
PRINT n5. 4000
XP t;4 = i;3 + v2
SP v7 = t;9 + 10
XP n3 = 1 + n3
X n2 =
S n4 = n7 - nl

CR LP + 345,6789 Sp Sp + 1 Sp Sp

Sp - 100.0000 Sp Sp + 6 Sp Sp

Sp + 12.3456 Sp + 12 Sp Sp

CR LP - 123.45

Sp + 9.876543

CR LP Sp Sp + 12

Sp - 7732

CR LP + 0. 12345678901 Sp Sp + 2 Sp Sp

Sp - 0.98765432109 Sp + 11 Sp Sp

CR LP Sp + 103

CR LP

205 -

9.3

SOME

PROGRAMMING

TECHNIQUES

There are a few other, miscellaneous, instructions in the Autocode. We shall describe all these
later except for one simple one; the instruction

STOP

causes a 77-stop. If the Run key is operated the next Autocode instruction will be obeyed.
The programme tape for an Autocode programme is headed as follows:

(name of programme)

J 1.0

The tape up to and including the J-directive is read in by the Initial Orders in the usual way. The
rest of the tape is then read in by the Autocode instruct ion- input routine (the Autocode itself must
have been put into the store previously, of course). The autocode instructions making up the programme
are then punched (as written) in their correct sequence after the J 1.0. They are read in iuid each one
is converted into a block of ordinary machine orders and numbers and stored (actually in B300 onwards).
At the end of the tape we must punch certain special symbols to cause the programme to be started. This
is done by brackets.

An instruction or a group of instructions written in brackets is obeyed as soon as it has been read
in. Since the first instruction in the programme is always automatically labelled we can enter the
programme (i.e. start obeying it) by punching

(-0)

at the end of the tape. This is an unconditional jump to the beginning of the programme, and since it
is written in brackets it is obeyed the moment it has been read in. If we like we can punch a sequence
such as the following on the tape:

(vl

- 1.76

= 10

This sequence is all read in but a note is made of the left bracket; when the right bracket is read
the instruction after the left bracket is obeyed, and then the next, and so on. The last instruction
is here an unconditional jump to the instruction labelled 3 and the main programme will be entered at
this point. A sequence of instructions like this is called a bracketed interlude. If the last
instruction of the interlude does not cause a jump then further instructions, punched after the interlude
on the tape, will be read in and will obliterate the interlude. If we wish, the bracketed interlude
can be quite a complicated prograimne including loops and print instructions, for example. In fact we
can even put the whole of our programme in brackets, when it will be entered as soon as it has been
read in; this technique is not recommended, however.

As an example of a complete Autocode programme we give below a programme to read in some numbers
from a tape in the main tape-reader and to evaluate and print the sum of their squares, and also the
square root of this number.

SUM OP SQUARES, SQUARE ROOT

J 1.0

STOP

v2 = TAPE *

vO =

2)i;l = v(l + nO) X t;(l + nO)

vO = vO + vl

nO = nO - 1
- 2, nO 7^

PRINT vO, 1025

vO = SORT vO

PRINT vO. 2025
STOP
(- 0)

wait for number tape

numbers to v2, v3, ... and set

nO = number of numbers

square t;(l + nO)
accumulate sum in vO
count numbers

print sura of squares

print square root

enter programme

206

SOME

PROGRAMMING TECHNIQUES

9.3

The instructions making up a programme are read into the store and converted at the rate of about
2 instructions per second, they are obeyed at the rate of about 15 per second. These figures are
necessarily approximate.

As another illustration we give an Autocode programme to tabulate the function

z = argsech y = - log [{l -/(I - y^)}/y].

for y = 0.01 (0.01) 0.99. We generate the current value of y by dividing an integer nl by 100; nl will

start at 1 and go up to 99. This is to prevent the accumulation of rounding errors. t The numbers y and

z are printed in two columns and an extra blank line is inserted (by printing CR LP) whenever y is a
multiple of 0.05 (i.e. n2 is a multiple of 5).

TABULATE ARGSECH - AUTOCODE

J 1.0

nl =

l)n2 =

nl * 5

- 2. n

2 7^

X n2

2)vl =

nl/100

vl, 3022

v2 =

vl X vl

v3 =

1 - v2

v4 -

SORT t;3

v5 =

1 - V4:

v6 =

t;5/t;l

vl =

-LOG v6

vl, 4027

nl =

nl + 1

- 1, nl 7^ 100

STOP

(- 0)

n2 is the remainder when nl is
divided by 5, if it is zero
we print CR LF

y = nl/100
print y

1 - y'

/(I - y2)
1 - /(I

y')

{l -/(I - y^)}/y

print z

enter the programme

As at present arranged there is room in the scheme for 594 Autocode instructions." This includes
the instructions in the last bracketed interlude (in such an interlude the right bracket occupies the
space of one instruction). Each instruction in fact occupies one block in the main store starting at
B300. (The starting block can, however, be altered if necessary.)

We shall now describe a few other facilities provided in the Autocode scheme which may occasionally
be useful. The instruction

TAPE

causes more instructions to be read in from the input tape (in the main tape -reader) and to be added at
the end of the programme (in fact they overwrite the last bracketed interlude). This instruction can
be used in a programme designed to do a rather complicated calculation on a few numbers or parameters.
At the end of this calculation a TAPE instruction can be used to read in a short bracketed interlude,
such as the following,

(n4 = 3
vl - 41.509
-^ 6)

This interlude can set new values of the numbers or paramete_:i and cause the calculation to be
repeated. *

A variant of the TAPE instruction is written, for example,

TAPE 6

t We could alternatively have used a variable ranging from 1 to 99 and divided it by 100 to give y.
This is also exact.

tt In the version of the Autocode for the 4096-word store there is room for only 210 instructions.

# The TAPE instruction should be carefully distinguished from the instructions such as f 1 = TAPE
and so on, which were described earlier; these latter read in numbers from the tape.

207

9.3 SOME PROGRAMMING TECHNIQUES

This reads in more Instructions from the input tape and puts the first one over (i.e. in place of) the
instruction labelled 6. Similarly the instruction

TAPE 6, 3

will place the first of the new instructions over the 3rd instruction after that labelled 6. We can
also use instructions such as

TAPE nl

TAPE nl,. n2

with similar effects.

It is sometimes desirable to stop obeying an Autocode programme and to start obeying ordinary
Pegasus machine orders; after some special calculation has been done we may then wish to return to
the Autocode programme, either at the next instruction or at some other specified instruction. There
are consequently facilities for exit to machine orders and entry from machine orders. The instruction

- M1682

means: start obeying machine orders at decimal address 1682 (i.e. at B210.2). The effect is similar
to a J-directive with decimal address 1682, so that B210 will be transferred to f/0. the next three
blocks to Ul. 2 and 3, and a jump will then occur to f/0. 2 (a-order). A link is set in XI and a
special word in X2; if the link is obeyed with C(2) undisturbed then the machine will return to the
next Autocode instruction (i.e. the one after the •-M1682). In this way we can jump to any a-order.
Similarly the instruction

- M1682, v2

will cause exit to machine orders at decimal address 1682, but in addition the variable v2 will be
unpacked - XI will contain its argument and X5 will contain 256 + its exponent. We can also use
instructions such as

- M n3, v5
or

- M1891. nl

This last will leave nl in 7^. With any - M instruction the settings of the accumulators will be
reproduced in SO. Entry from machine orders to an Autocode programme at the instruction labelled 15
(for example) can be done by putting 15 into X2 as an Integer and obeying the cue

37 [372

4.4 60

Before obeying this cue we can set a link in XI; this is obeyed by the Autocode instruction

- L

If we wish to use machine orders together with an Autocode programme we have to know how the space in
the main store is allocated. This is at present as follows: -

Bl to 111 Autocode scheme

5111.2 onwards Labels (see below)

fil24.0 to 127.3 Indices (nO to n27 - one location each)

B127.4 to 299.7 Variables (vO to j;1379 - one location each)

B300 onwards Instructions (one block each)

The labels each occupy one location, thus the word in Bill, 4 gives the address of the instruction
labelled 2. If labels to 25 are used, then locations B113.2 to 114.3 must not be used for machine
orders. This allocation of the store is liable to alteration if extra facilities are included. It is
probably safest to use for machine orders the space normally occupied by the higher-numbered variables.
The Initial Orders may be called in as a subroutine by the Autocode instruction

- 10

A link is set which causes return to the next Autocode instruction when an L-directive is read,
provided the contents of BO. 1 and 0. 2 are undisturbed. Similarly the instruction

- 10, 1200

calls in the Initial Orders after setting the Transfer Address to decimal address 1200 (i.e. to
jB150. 0), and the instruction

- 10, 1200, 153

208

SOME PROGRAMMING TECHNIQUES

9.4

▲

will in addition set the relativizer to 153. If we wish we can also use instructions such as the
followi.ng.

- 10, nl

- ID, nl. n2

These various Autocode instructions are summarized in Table 9.6

STOP
TAPE
TAPE nl
TAPE nl. n2

- M nl

- M nl, v2
~* M nl, n2

- L

- 10

- 10, nl

- 10. nl. n2

Stop (77). proceed with next instruction when Run key is
operated.

Read in more instructions and add them at the end of the
programme.

Read in more instructions, the first to replace that
labelled nl.

Read in more instructions, the first to replace the
instruction n2 after that labelled nl.

Set
link
in XI

Exit to machine orders at decimal address nl.

The same, and leave v2 unpacked in Xl and X5.

The same, but leave n2 in 7^

Obey link set in XI before entry from machine orders by cue.

Call in Initial Orders as a subroutine.

The same, but set T.A. = nl.

The same, and set relativizer = n2 in addition.

Table 9.6 Autocode instructions

liscel laneous

At the beginning of the Autocode programme tape the directive J 1.0 is punched; this causes the
input programme to treat the following instructions as a new Autocode programme and to assign the label
to its first instruction. Sometimes we may wish to restart an Autocode programme that has already
been read in, perhaps after changing one or two parameters or after making some alterations to the
programme (this is described below). This can be done by reading in a bracketed interlude ending with
a jump, for example

(n3 = -48
- 1)

We must not punch J 1.0 at the beginning of this tape for the input programme would then treat it as
the start of a new programme and store it over the beginning of the original programme. We must
instead punch J 1.2 at the head of the tape; this has the same effect as the Autocode instruction TAPE
so that any instructions on the tape will overwrite the last bracketed interlude.

Occasionally it may be found desirable to alter an instruction in an Autocode programme. Ttiis
can be done by using the tape-editing eQuipment to alter the tape but this may be laborious if the
programme tape is long. Instead of doing this we can insert a correction near the end of the tape,
before the bracketed interlude causing entry to the programme. This correction is written, for example

ALTER 6, 3

followed by an Autocode instruction (on a new line); this instruction will then replace the one which
is three after the instruction labelled 6. This sequence forms a kind of Autocode directive and is not
stored; it does not form a part of the programme. When the alteration has been made the input section
of the Autocode scheme will return to read more instructions or alterations. If the alteration is to
be made by a separate tape then it should be headed J 1.2 as described above. In this connection it is
worth noting that a bracketed interlude can be considered as a kind of Autocode directive.

V 9.4 The Matrix Intei-pretive Scheme

Many of the calculations arising in industry can be expressed in matrix form. '^^ It is therefore
important that the preparation of these calculations for the computer should be as quick and easy as
possible. The matrix Interpretive Scheme is a means of specifying in easy and concise terms the
transformation between the matrix form of the problem on paper and the actual operation of the machine.
The problem to be solved must first be expressed in matrix form. Then a method of solution has to be
thought out and expressed as a sequence of elementary matrix operations, e.g. the input of a data-matrix
into the computer, or the multiplication of one matrix by another. Each of these elementary steps is
then written down as a single instruction, called a matrix- ins traction, which defines the operation
required and the positions in the computer' s store of the matrices involved. The whole sequence of
matrix-instructions is called a matrix-programme.

t Good introductions to Matrix Algebra will be found in the following books. AITKEN, A. C. , Determinants
and Matrices (Oliver and Boyd, Edinburgh, 19). PERRAR, W,L. , Algebra (Oxford university Press).
There is also a useful discussion of some of the applications in: PRAZER, R. A. , DUNCAN, W.I. and
COLLAR, A. R. , Elementary Matrices (Cambridge University Press, 1938). LANCZOS, C. , Applied Analysis
(Pitman, London, 1957).

209

9.4 SOME PROGRAMMING TECHNIQUES

In use the programme of the Matrix Interpretive Scheme is first read into the store of the
computer by the Initial Orders in the ordinary way. t The Interpretive Scheme programme then reads
in the tape on which is punched the matrix -programme. When the whole tape has been read and stored the
individual matrix-instructions making up the matrix -programme are examined and decoded in turn by the
Interpretive Scheme programme, which then carries out the operations called for. The notation used for
the matrix- Instruct ions has been designed to be simple. The aim has been to make it possible for the
person originating the problem to write the programme in this form, which is directly acceptable by
the computer; the user need not be familar with ordinary Pegasus programming.

The user of the scheme can visualise the store of the computer as a continuous strip of 6142
locations, each of which can hold one element of a matrix. ^^ 'Hie elements of a matrix occupy a block of
consecutive locations on this strip. It is an important practical point that a stored matrix consists
of nothing more than its individual elements arranged in a certain order, and that each element can
stand on its own as a number. Thus there are no row or column checksums or overall scale factors, and
the dimensions of the matrix are not stored explicitly with it. This makes it possible to change or
extract individual elements, or groups of elements, to regard a rectangular matrix as a number of columns,
to deal relatively easily with partitioned matrices, and in general to make efficient use of the storage
space available. When a particular matrix is no longer needed the space it occupies in the store may
be used again at a later stage, giving great economy of storage space.

An Important feature of the scheme is that the user can consider all numbers to be in their normal
decimal form (with the decimal point wherever he wishes); when these numbers are taken into the computer
they are converted automatically into packed floating-point numbers (4.2"). which are used for all
subsequent operations (the converse applies on output). This eliminates difficulties in scaling or
overflow and enables the numerical data to be read into the machine in the form in which they occur.
All the numbers are represented with a precision corresponding to the use of between eight and nine
significant figures; this is adequate for most applications.

The instructions and the numerical data are punched on to paper tape. The elements of the matrices
are punched as written, each preceded by its sign and followed (optionally) by a decimal exponent. Thus
an element of value 123.456 may be punched in any of the following ways: -

+ 123.456

+ 0.123456 + 3

+ 1.23456 + 2

+ 123456 - 3 (the spaces are optional).

Each element may be labelled (if desired) by its row or column number, punched as a number without a
sign.

Matrices are normally punched by columns (this giving fewer errors) and after each element the CR
and LP characters are punched. This permits a data tape to be typed out automatically by teleprinter
equipment for checking or record purposes. It is normal to punch a checksum at the end of each column;
this is used by the computer to check the input when the tape is read in. It should be noted that this
checksum is not stored, it is merely compared with the sum of the elements read in, and input carries on
as usual if the check agrees. Letters, numbers or other characters may be punched at the head of the
tape as a name or label. When the tape is read in these will be punched on the output tape, which thus
bears a record of the matrices which have been put in.

Results are normally presented in columns, in a form similar to that used for input. Each column
is headed by its column number, each element is preceded by its row number, and there is a checksum
for each column. By using scissors and paste the result can be displayed as a complete matrix in its
conventional form. There are two alternative forms of output:

(a) each element has one decimal figure before the point and is followed by a decimal exponent;
or (b) the whole matrix is automatically scaled (by a power of 10) so as to print the largest element
with a pre-selected number of figures before its decimal point, and the decimal exponent of
the scaling factor is printed at the head of the matrix.
The punched tape which appears as the output may be printed out automatically; it is in such a form
that it may be read in again if it is being used to carry intermediate results.

In the interests of simplicity no checks on the accuracy of computation have been incorporated.
Experience with the computer has shown that there is no real need for these, and in any case they
would require a considerable amount of rather difficult programming. However, the computer itself
incorporates a number of automatic checks against errors, particularly concerning the input, output
and the store. There is a progranmied check against errors in the input equipment. The column checksums
will help detect wrongly-punched data. During the input of any matrix the elements are counted and
checked against the dimensions of the matrix.

The matrix interpretive scheme enables a complete matrix operation to be performed by writing one
single matrix-instruction. To enable a complete matrix-programme to be built up it is necessary to
have certain other functions. These have been provided for in the scheme and a complete matrix-programme,
comprising many separate matrix- instruct ions, can be built up quickly and easily. The list of
functions is comprehensive for most matrix work. A single matrix can be copied, transposed or
normalised. Two matrices can be added, subtracted, multiplied or divided. Matrix-instructions are also
used for the input and output of matrices. In addition to rectangular matrices, the scheme is designed
to operate on diagonal matrices, row-vectors, column -vectors, and scalar matrices. The row- and column-
vectors are regarded simply as special cases of rectangular matrices.

t The scheme bears the Library routine-number R 7500 (or R 2500 for the 4096-word store), which is used
only for reference purposes.

tt On the 4096-word store there is room for 3070 matrix elements.

210

SOME

PROGRAMMING

TECHNIQUES

9.4

Each matrix element stored in the computer occupies one storage location, of which 6, 142 are
available. A matrix is stored within this space in columns; a matrix of order m x n occupies mn
storage locations. For a diagonal matrix, only the diagonal line is stored, so that a diagonal matrix
of order n x n occupies n storage locations. A scalar or a scalar matrix (i.e. a scalar multiple of
the unit matrix) is stored as a single element. The scheme has been written so that in most operations
the results may be written over one or other of the operands; exact details of what may be done are
given later.

The following is a typical matrix-instruction:

(304, 10 X 12) X (1195. 12 x 16) - 5

When obeyed this will cause the 10 x 12 matrix starting at location^ 304 to be post-multiplied by the
12 x 16 matrix starting at 1195, the product will be a 'matrix of 10 rows and 16 columns, and this is
placed in consecutive locations starting at 5. The whole operation is done in floating-point form.
Each expression in brackets on the left can be thought of as representing a matrix. A matrix may be
rectangular, for example (2107, 29 x 25) stands for a matrix with 29 rows and 25 columns whose first
element is in location 2107; this matrix will occupy 725 locations of which the last is 2831. A
diagonal matrix with 19 rows and columns starting at location 100 is written (100, 19/); since only
the diagonal elements are stored this matrix occupies only 19 locations. No order need be specified
for a scalar quantity or a scalar matrix; for example (300) can mean a scalar stored in location 300
or a scalar multiple of a unit matrix of any order.

In a general instruction three matrices may occur; the operands will be referred to as A and B
and the result as C. The addresses of these matrices will be denoted by Na, Nf,, N^, respectively
(these are the addresses of the locations holding the first elements of the matrices). The letters
m and n denote the number of rows and columns respectively. In general the following three kinds of
matrix are used.

(a) A rectangular matrix is written {N^^^, m x n) ,

(b) A diagonal matrix is written {Na, W),

Row- and column-vectors are regarded as special cases of rectangular matrices, for example (1230,
12 x 1) means a column-vector with 12 elements. In any matrix. N, the address, must lie between 1 and
6142, and the dimensions m and n must not exceed 255. The address A^ is zero in certain orders used
for input and output.

The available types of matrix-instruction are shown in Table 9.7; for each one a general symbolic
form is given, and also a typical actual matrix- instruct ion. Apart from a few obvious exceptions, any
matrix may be rectangular, diagonal or scalar. In the symbolic form. A, B, C represent the three matrices
concerned; A and B are written in the way described above and C is specified merely by its address.
There are two instructions which are not included in this table (written with letters and J); these
are described later.

Function

General form

Typical matrix- instruct ion

Copy

k-* N

(314, 15 x 9) - 2653

Input

- A'
c

(0, 58 X 9) - 932

Output

A {x. y) - (384, 6' x 2) (6, 4) -
and y specify the number of digits - see below)

Transpose

A * - /V
c

(33, 8 X 32) * -* 795

Transpose in situ

(A I

A/ ->iV^
tiust be square, C must have

(348. 25 X 25)/ - 348
the same address as A)

Add

A + B - A^
c

(28, 8/) + (41, 8 X 8) - 971

Subtract

A - B -> A^
c

(6, 9 X 3) - (99, 9 X 3) - 751

Multiply

A x B - A^^

(375, 10 X 5) X (58, 5/) - 209

Divide

(in this or

A, B - N^
der A" ^B goes toN^, A and B

(749, 4X4), (592, 4 x 2) ^ 30 '
are both in general spoiled)

Normalize A n B - iV^ (78. 16 x 40) n (62) - 862
(a scaled form of the matrix A goes to N g. the largest element being
made 1, the scaling factor, which must be a scalar, goes to N},)

Convert to
fixed-point

(the scale
the maximu

A v B - iV^
factor goes to /V^ ; this i£
m element)

(8. 9 X 9) J^ (862) - 901
i the exponent of

Binary output

A V - 1 (or 9)

(34, 21 X 17) V - 1

Binary input

(these last

v - Af^
two functions are complemen

(0, 6 X 7) u - 982
tary; they are described below)

Stop (77)

♦

Table 9.7 Matrix-instructions

t The 6142 available locations (or 3070 on the 4096-word store) are numbered straight through (starting
at I) , these numbers are not the ordinary decimal addresses of the locations but differ from them only by
addition of a constant (actually 1023).

211

9.4 SOME PROGRAMMING TECHNIQUES

As an example, we give below a complete programme for the calculation of the column-vector y given

y = {con + Q(E + fl) b}x,

where oj, Q and / are scalars, D is a diagonal matrix, E and B are square matrices, I is the unit matrix,
X is a column-vector, and all matrices and vectors are of order 10. For checking purposes, the last row
and column of the intermediate square matrix

C - Q(E + /I) B

are to be printed. The data for the calculation are

E, B, /, U CO, and x

and are assumed to be available on an input tape in this order. The complete programme tape reads as
follows: -

SPECIMEN MATRIX -PROGRAMME

J64.0

Date and serial number
name of programme

Call in matrix scheme

(0, 10 X 10) - 1

(0, 10 X 10) - 101

(0) - 201

(0) -• 202

(1, 10 X 10) + (201) - 1

(202) X (1, 10 X 10) - 1

(1, 10 X 10) X (101, 10 X 10) - 203

(203, 10 X 10) * - 101

(191, 10 XI) (6) -

(293, 10 X 1) (6) -

(0, 10/) - 101

(0) - 202

(101, 10/) X (202) - 101

(101, 10/) + (203, 10 X 10) -> 203

(0, 10 X 1) - 1

(203, 10 X 10) X (1, 10 X 1) - 11

(11, 10 X 1) (6) -

Read in E
Read in B
Read in /
Read in 0.
Replace E by (E + /I) , and

then by H (E + /I)
Form C

Replace B by c' (transpose of C)
Print out last column of C
Print out last colujnn of C
Replace c' by D (diagonal)
Replace D. hy co
Replace D by Ci)D
Replace C by coU + C
Read in x

Form y = (coD + C) x
Print out y
* Stop

S Directive to start obeying

mat r ix - pr ogr amme

To run this programme the tape of the Matrix Interpretive Scheme is first run into the computer; it
ends with a Z-directive to cause a stop. The above tape is then put into the main tape-reader and the
Run key is operated. The D and N-directives have their usual effects and the special directive J 64.0
is then read; this simply causes the Matrix Interpretive Scheme to read in the following programme
and store it. The S at the end of the tape is not a matrix-instruction but a directive (S for start)
which causes the matrix -programme to be obeyed. The first order is, as usual, an input instruction;
this kind of instruction always causes an optional stop before any tape is read (to allow changing of
tapes, if required). At this point the data tape is placed in the main tape-reader and the Run key is
operated. The actual running time for the above programme, including input of data and all output, is
about 75 seconds.

The instructions making up a matrix -programme are, of coui'se, punched exactly as they are written,
each being terminated by CR and LF. It is usual to include a number of spaces in each instruction to
improve the legibility of the print-out, these spaces (as well as any Erase characters) are ignored when
the tape is read. There is room in the store for 80 matrix -instruct ions, these being numbered to 79.
Should the matrix -programme be longer than this, one can either use some of the space normally used
for numbers or else read in more instructions after the first 80 have been obeyed. To assist in
reading in the matrix -programme three directives have been provided in the order-read section of the
Matrix Interpretive Scheme; these are as follows.

I This causes the Initial Orders to be called in (as a subroutine). When the Initial Orders read
an L-directive the interpretive scheme will resume reading in matrix-instructions at the
point it had reached when the I was read.
S This causes the interpretive scheme to start obeying the matrix- programme. If desired we can
write a number after the S; for example, S 12 causes the scheme to start at the matrix-
instruction no. 12 (note that the instructions are numbered from 0).

212

SOME PROGRAMMING TECHNIQUES 9.4

Xn This directive sets an address (the equivalent of the Transfer Address) in the scheme so that
the next matrix- instruct ion on the tape will replace matrix-instruction number n. If n is
zero it may be omitted.
The numerical data read in by input matrix -instruct ions can be punched as normally written. Each
number can be punched with or without a decimal exponent. A number is introduced by its sign (punched
as + or -); spaces and erases are ignored throughout the number. Tlie decimal point is punched where
required; no decimal point need be punched If the number is an integer. The number of decimal digits
which can be accommodated in one number is 9. If more than this are punched they will be ignored,
but care should be taken not to punch more than 11 digits if the number is being punched as an integer.
If the number is fractional then the extra digits will in any case be insignificant. Tlie exponent (if
there is one) is punched after the number proper and is introduced by its sign; it must not exceed 77.
Tlie end of the number is indicated by CR LF. If desired, unsigned integers representing row or column
numbers may be punched in front of any number, they will be ignored on input. The data- input section
of the matrix scheme recognises the following directives.

n This is punched to introduce a decimal exponent (a block exponent) which is to be carried through
the matrix. It is followed immediately by a sign and then the exponent. This will be added
to the exponent of each number until an asterisk or another n is read. If no exponent is put
in this nuitiber will of course be zero. Space and erase are ignored. The exponent is
terminated by CR LP.
~* It is possible to print out a name or title for each matrix. This is done by punching an - on
the data tape. Each character read after this will be printed on the output tape until two
consecutive figure shifts are encountered.
= This is punched immediately before a checksum. Thereafter the checksum is punched in the same

way as any other number. A certain tolerance is allowed between the checksum computed from the
elements, and the checksum on the tape; this tolerance depends on the number of digits actually
punched in the tape checksum. If the checksum is not exact, i.e. if the elements of which it is
the sum have been rounded off after computing the checksum, it should be punched to as many
digits as are punched in the element which is punched to the fewest number of digits (disregard-
ing zero elements). If however the checksum is intended to be exact, as is more common in
practice, it should be punched with two more digits than are given in the separate elements,
subject to one restriction. The restriction is that the sum of the moduli of the checksum and
the largest element should not contain more than 9 digits. This principle can best be illu-
strated by examples: -

(a) +.2031

+ .101
+ .123

Sum +.4271
Punch = +.427100

(b) +10210

+320

Sum +10530

Punch = +10530.00

+.3641259
-1000

Sum +23.3641259
Punch = +23.36413

(d) -.12345

+.12345
Sum +.00000
Punch =^ +.0000000

(If +0 is punched, even such a gross error as the wrong sign for either element will not be
detected. )
* This must be punched after each matrix. It causes the next matrix-instruction to be obeyed,

after first checking that the correct number of elements has been taken in.
There are two programmes within the scheme for the output of results; they are independent of one
another, and the scheme has been written so that only one may be in the machine at any one time. One
programme prints numbers in floating-point form, that is, with one decimal digit preceding the decimal
point and the whole argument followed by an exponent to base 10. The other programme prints numbers in
a fixed-point form with a specified number of digits before the decimal point and a block exponent for
the whole matrix (which is printed first). There are thus two matrix-schemes, identical except for
form of output, and they are known as "floating-point matrix scheme" and "fixed-point matrix scheme".

A tape produced by the floating-point output programme may later be read in by the input programme
(provided that not more than 9 significant figures have been punched). A tape produced by the fixed-
point output programme can also normally be read in again, but if this is to be done not more than 8
figures should be punched in each number (if 9 are punched there may be occasional checksum failures on
input) .

- 213

9-4 SOME PROGRAMMING TECHNIQUES

A name or title is printed to distinguish each matrix which is punched out. It consists of - and
a decimal number specifying which matrix instruction caused output to occur. For example, -16 means that
the 16*^ instruction is being obeyed. The elements of the matrix are printed in columns. Each column
is preceded by two line feeds and the column number, which appears centrally over the column. Each
element is preceded by a row number and two spaces. These may be suppressed at any time during punching
by pressing down the sign-digit key of the handswitches.

If the floating-point output is used each number is printed as an argument and a decimal exponent.
The number of significant digits in the printed argument is x, the number written in the output instruc-
tion (if none is written x is automatically set to 9). A typical element with x = 7 is

12 +1.234567 +16

meaning 1.234567 x 10^'^; the 12 is the optionally printed row number. At the foot of each column a
checksum is printed, preceded by an extra LF and an equals sign. The number of figures printed in this
checksum is automatically adjusted to suit the tests applied on re-input.

If we use fixed-point output each number is printed with at most x significant figures, of which
at most y appear before the decimal point (there will always be x-y digits after the point); here x
and y are written in the output instruction. (If x and y are not written in this instruction they will
be set to 9 and 1 respectively). Two elements might appear as follows, if % = 8 and y = 4

15 -8765.4321

16 +45.6789

provided the row numbers are not suppressed. A checksum is printed at the foot of each column.
Pig. 9.1 shows specimens of the two kinds of output. The fixed-point output includes on overall scale-
factor or block exponent (printed at the head of the matrix); the programme can easily be amended to
give a normal fixed-point output with no scale factor.

n+2

+1,3742

+ 2» 1840

+ 2

"1*933^

+ 2

-6.001 1

+ 3o6 100

~I0

+7 #9 3 2 +2

+2«6o98

-I

-I • 1122

+ 1 . 1132

+5.3142

-2

+4.1 592

+ 1

+4. 1907

+ 1

+ 2. 184

2
3

-7*934
~o »o6o

+0 .000

+7«932

+ .CO 2

I
2

3 ;

-o.or I

+0.01 ^

"^0 »ooi
+0 .416

+0.419

-8.9793 +3 o -89,793

1 -^o I +0,000

2 -2.3846 -I 2 -0.002

3 -2.6433 +1 3 -0,264

4 +8,3279 +3 4 +8.328

= -8.1732 +3 = -81,732

Pig. 9.1 A typical matrix with 5 rows and 3 columns as printed by the Matrix Interpretive
Scheme, using the two alternative forms of output.

214

SOME PROGRAMMING TECHNIQUES 9.4

The only instructions not so far described are those for exit to machine orders (written as a
letter followed by the decimal address of an a-order) and for jumping; there is only an unconditional
jump, written as a J followed by the number of the matrix -instruct ion to which the jump is to go. The
0-instruction causes the block containing the specified a-order to be transferred to UO and entered;
5j^ will be the address of this order and may be used to read in more blocks from the main store. To
return to the next matrix- instruct ion we must obey the cue

61 [o]72

0.0 60

To return to matrix- instruct ion number n we put the integer n into XI and obey the cue

65 (T|72

1.2+ 60

The 0-lnstructlon is useful if a matrix -programme is very long and there is room only for a part of it
in the store; at the end of each part we may have the instruction 512 to read more matrix -programme
over the top of the old, or 7168 to enter the Initial Orders. ^ The Matrix Scheme occupies blocks up
to jB107 inclusive, and the matrix -programme starts at B108.0 (with two words per matrix-instruction);
care must be taken not to overwrite these by machine orders. If there are only 60 matrix- instructions
in the programme then B123 to B127 inclusive are available for machine orders, or we can use part of the
space normally holding numbers. Each matrix element occupies one word and location 1 corresponds to
B128.0.

There is an extension to the scheme to permit the use of preset-parameters. For a problem of a
given type this allows one matrix-programme to be written which can be used with matrices of different
sizes. Vfhen this facility is to be used the programme is written in general terms, and a small
subsidiary programme (called the Interlude) is used to insert into the matrix- instruct ions the dimensions
of all the matrices for the particular calculation, and also to allocate the storage locations. To use
this facility we write the matrix -programme in the ordinary way except that any dimensions (i.e.
numbers of rows and columns) which cannot be fixed are written in as numbers between 241 and 255. and
any addresses which cannot be fixed are specified as numbers greater than 5000. In fact if m denotes
any dimension and N any address of a matrix as written in an instruction, then

(a) if 1 < m ^ 240 m is the actual dimension, but

if 241 < m < 255 the dimension to be used is specified

separately by the programmer,

and (b) if 1 ^ iV < 5000 N is the actual address, but

if iV > 5001 the address to be used is specified

separately by the programmer.

While writing the matrix -programme we build up two lists, an address-list containing values of N and a
dimension- list containing values of m. In the actual instructions we write 5001, 5002, etc. for
unspecified values of N, and 241, 242, etc. for m. For a particular case we can punch actual address
and dimension lists to suit the matrices being used and we feed these into the store together with the
matrix -programme. The Interlude then processes the matrix-prograirane. replacing any N exceeding 5000
and any dimension exceeding 240 by the appropriate address or dimension from the lists. After this the
resulting matrix-programme is obeyed in the usual way. The advantage of the preset-parameter scheme
is that the actual matrix -programme will apply to all problems of a given type, and yet we retain the
possibility of overwriting matrices when they are no longer needed.

As an example, we give a short matrix-programme to read in two matrices

Q, of order m ^ n (m ^ n) ,
and A, of order m x m,

and to evaluate and print the n x n matrix Q'AQ This programme is written as follows.

(0, 241 X 242) -* 1 Read in Q (m x n)

(0, 241 X 241) - 5001 Read in A (m x m)

(5001, 241 X 241) X (1, 241 x 242) - 5002 Form AQ (m x n)

(1, 241 X 242) * - 5001 Form ft' (n x m)

(5001, 242 X 241) - 1 Copy Q' over Q

(1, 242 X 241) X (5002. 241 x 242) - 5001 Form q'AQ (n x n)

(5001, 242 X 242) - Print result

* Stop

t The instruction 4096 is used to enter the Initial Orders in the 4096-word store version.

215

9.4 SOME PROGRAMMING TECHNIQUES

We prepare the dimension and address lists in the following form while this programme is being written.

Dimension list Address list

241 m 5001 2mn + 1

242 n 5002 mn + 1

The allocation of the store is as follows:

1 to mn Matrix Q, later ft'

mn + 1 to 2mn Product AQ

2 mn + 1 to 2mn + m^ Matrix A

2mn + 1 to 3mn Matrix Q' (overwrites A)

2mn + 1 to 2mn + n^ Matrix Q'aQ (overwrites A and Q,')

The programme can be used for any matrices A and Q provided

2mn + n^ ^ 6142.

In use actual numbers are put into the dimension and address lists, and these are then read into
decimal address 1600 and 2000 respectively (i.e. into B200.0 and B250.0); this preset -parameter
tape is made up as follows (for the above matrix-programme with m - 30, n = 31, 2mn + 1 = 1861,
mn + 1 = 931):-

T 1600

+ 30

+ 31

T 2000

+ 1861

+ 931

J 40.0 entry to Interlude.

dimension-list

address-list

This tape is to be read in after the matrix-programme has been read in as usual. At the end of the
matrix-programme tape we should punch the Matrix Scheme directive I to call in the Initial Orders
(there can then be an Initial Orders Z-directive to cause a stop so that we can put the parameter tape
in the tape-reader). When the Interlude has been entered by the J 40.0 at the end of the parameter-
tape it first finds the address of the last matrix-instruction read in and then processes all the
matrix- instructions, replacing any addresses greater than 5000 or dimensions greater than 240 by
numbers taken from the lists. When it has done this the computer will start to obey matrix-instructions
starting at number 0.

If a programme like the above example is used a great deal we can arrange to feed in only the
dimension-list and get the address-list automatically computed by a small programme of machine orders.

If storage space for matrices is at a premium it may be desirable to punch out certain intermediate
results for later re-input. This is best done by using the binary input and output instructions. To
punch out in binary the 10 x 6 matrix in location 100 we can use the instruction

(100, 10 X 6)v - 1

which precedes the main punching by a leader of blank tape; a checksum is punched at the end of the
matrix. To read this matrix in again to location 947 we use the instruction

(0. 10 X 6)t; - 947

If we use the following instruction to punch out the matrix

(100, 10 X 6)v -' 9

a checksum is punched at the end of each column and the matrix can be read in either one column at a
time, or as a complete matrix. If desired we can punch out the successive columns of a matrix (e.g.
as they are computed) by several Ov -* i instructions and later read it in as a complete matrix.

In most matrix- instructions the address of the result (C) can be the same as that of either of
the operands (A and B) , provided the result does not occupy more storage space than the operand which
gets overwritten. The following are the exceptions to this rule.

(a) Transposit ion

\ * -* ^' C may never overwrite A,

A / - /V C must always have the same address as A.

(b) Multipl ication

A X B - yv

If either operand is rectangular (but not a vector) then the other one may not be overwritten.

216

SOME

PROGRAMMING TECHNIQUES

9.4

A, B - Ncii.e. C = A" ^ B)

If A Is square it may not be overwritten; if in addition B is rectangular (but not a vector)
it may not be overwritten either.

In the division instruction A is spoiled if it is square, and is replaced by A- 1 otherwise (unless
overwritten). If A is square, B is replaced by c' (if B is rectangular) or by C (if it is a column-
vector); in all other cases B is untouched provided, of course, it is not overwritten by the result.
Note that partial overwriting of an operand is forbidden in general for any instruction.

There are a few useful tricks which can be done with the transpose instruction A * -* C; in fact
an order of this kind is fundamental to any scheme for manipulating matrices. It is easy to pick out
the individual columns of a matrix; by transposing first we can pick out the rows. This idea can be
generalised to partitioned matrices. Suppose a 10 x 8 matrix has address 1 and is partitioned thus

where A is 4 x 5, B is 4 x 3, C is 6 x 5 and D is 6 x 3. This matrix occupies locations 1 to 80.

Suppose we have to put the component matrices A, B, C, D into addresses 81, 101, 113 and 143 respectively.

The following sequence of matrix- instruct ions can be used:

(1. 10 x 5)* - 123
(123, 5 x 4)* - 81
(143, 5 X 6)* - 113
(51. 10 X 3)* - 149
(149, 3 X 4)* - 101
(161, 3 X 6)* - 143

Matrix (A' I C') to 123-172

A to 81-100

C to 113-142

Matrix (B' j d') to 149-178

B to 101-112

D to 143-160

A similar technique can be used to assemble a partitioned matrix. The transpose instruction can also
be used to pick out the diagonal elements of a matrix. Suppose a 4 x 4 matrix A is at address 1. The
following instruction puts into locations 21 to 24 the diagonal elements of A (and incidentally over-
writes locations 25 to 40):

(1, 5 X 4)'

The diagonal matrix (21, 4/) is now made up of the diagonal elements of A.

The speed of operation can be found from Table 9.8. The approximate time for obeying an order
can be determined from the formulae given. The last three columns give some typical times for square
matrices and vectors of various orders. The bulk of the calculation time in most problems is likely
to be occupied by multiplications of one rectangular matrix by another or by non-trivial divisions
(i.e. those in which A is square); output time may also be significant. All the times given are
approximate because the time taken by a floating-point operation depends on the actual numbers involved
and because the access time to the main store cannot be accurately forecast. Usually the formulae give
times within about 10% of the actual times. In this table the following abbreviations are used:

Rect. Rectangular matrix,

Diag. Diagonal matrix.

Col. vector Column-vector,

Scalar Scalar or scalar matrix.

The elements making up a matrix are each packed floating-point numbers similar to those described
in Sec. 9.1 (with 30 bits for the argument and 9 bits for the exponent + 256). When used in the
Matrix Scheme, however, these numbers are only partially unpacked when operated on, in fact the whole
of the 39-bit word is used as the argument. The 9 bits used for the exponent are included and are
used to round the argument; the resulting bias is normally quite negligible (because the exponent is
stored augumented by 256) unless the elements are known exactly (e.g. if they are integers). Underflow
(i.e. an exponent becoming less than -256) causes the number to be replaced by zero; there is no
check on floating-point overflow, but this, of course, is unlikely to occur.

Each matrix -instruct ion occupies two words, the first of which must have an even address; the
digits in these words are used as follows:

First (even) word

13 bits

iVfe

Second (odd) word

not used

flifc

217

9.4

SOME PROGRAMMING TECHNIQUES

Here the figures show the number of binary digits used for each part of the instruction; these parts
are as follows:

Nat N(,, Nc the three addresses in the instruction,

F six bits indicating the function,

/»(,, n^ the number of rows and columns in the first matrix,

mj,, nj, the number of rows and columns in the second matrix.

Operation

Formula giving time in
milliseconds

Examples of times in seconds

A matrix

B matrix

COPY

Vector or Diag.

5m + 150

0.2

0.3

0,3

Rect.

5mn +150

1.4

2.9

5.3

TRANSPOSE

Ordinary

12mn + 130

3.2

7.0

12.4

'In situ'

13n2 + 150

3.5

7.6

13.4

ADD or SUBTRACT

Rect.

16mn + 190

4.3

9.4

16.6

Diag.
Vector

ll6m + 190

0.5

0.6

0.7

Square

Diag. )
Scalar J

(Diag.
[Scalar

■5m 2 + 21m + 200

1.7

3.6

6.0

Square

Diag.
Scalar

Scalar
Diag.

.15m + 190
J

0.4

0.6

0.7

MULTIPLY (all these times will be reduced if there are zeros

in A or B)

Rect. (mXn)

Rect. (nxr)

L(21n + 12)r + 12j m + 150

Im 29.4

4m 57.7

11m 41.0

Rect,

Col. vector

21mn + 24m + 150

5.9

12.8

22.4

Row vector

Rect.

21mn + 12n + 170

5.7

12.5

22.0

Row vector

Col. vector

2in + 180

0.5

0.7

0.9

Col. vector

Row vector

33mn + 12m + 150

8.8

19.4

34.3

Rect.

Diag.

lOmn + lln + 170

2.9

6.2

10.8

Row vector

Diag.

21n + 170

0.5

0.7

0.9

Rect.

Scalar

lOmn + n + 180

2.8

6.0

10.5

Row vector

Scalar

lln + 180

0.3

0.4

0.5

Diag. \
Scalar J

Rect.

12mn + lln + 170

3,4

7.4

12.8

Diag. "1
Scalar J

Col. vector

12m + 180

0.4

0.5

0.6

Diag.

12m + 180

0.4

0.5

0.6

Scalar
Diag.

Diag,
Scalar

I 11m + 180

0.4

0.5

0.6

DIVIDE

Square (nXn)

Rect. (nXr)

19n2r + 8.5n^ + 30nr + ISln^
+ 181n + 170

2m 37.0

7m 57.0

17m 52.0

Square

Col. vector

8.5n3 + ISOn^ + 211n + 170

Im 17.0

3m 29.0

7m 19.0

Diag.

Rect.

12nr + lln + llr + 170

3.6

7.7

13.2

Diag.

27n ■)- 180

0.6

0.8

1.0

Scalar

Rect.

12nr + llr + 180

3.4

7.4

12.8

NORMALIZE

15mn + 190

4.0

8.8

15.6

CONVERT

12Bn + 170

3.3

7.1

12.5

INPUT (the specimen times assume 6 digits, sign and point ir

1 each number

, and that t

here

are no checksums or row- or column-numbers).

Rect.

5 msec, per character read,

33.3

Im 14.9

2m 13.1

Vector

including layout characters,
signs etc. plus 70-90 msec, per
number

2.1

3.1

4.2

OUTPUT (the specimen times assume 6 digits etc. as for the i

nput times).

^^""l' \ Fixed-point
Vector J

Rect. I Floating-
Vector J point

30 msec, per character punched

Ira 43.0

3m 43.5

6m 30.1

including layout characters
etc. plus about 60 msec, per

6.7
2m 14.6

9.6

4m 55.4

12.5
8ffl 36,7

number, including checksums

8.7

12.5

16.3

BINARY INPUT

40mn

10.2

23.0

41.0

BINARY OUTPUT

250mn

Im 4.0

2m 24,0

4m 16.0

Table 9.8 Times of operation of matrix- instructions.

- 218 -

SOME PROGRAMMING TECHNIQUES 9.5

The function jF* indicates the kind of operation and the nature of the operands; the following are some
examples:

Function

f Operation

Transpose,

2 Multiply rectangular matrix by rectangular matrix,

6 Multiply diagonal matrix by rectangular matrix,

7 Multiply scalar matrix by rectangular matrix,
10 Add rectangular matrix and diagonal matrix,
12 Add diagonal matrix and rectangular matrix,
22 Subtract scalar matrix from diagonal matrix,
28 Stop,

38 Divide scalar matrix by square matrix.

This function is evaluated during input of the matrix-instruction; it is later used to cause entry to
the appropriate subroutine for carrying out the actual matrix operation.

If the storage space provided by the Matrix Interpretive Scheme is not enough to hold all the
matrices required we may be able to punch out (in binary) some intermediate results which can later be
read in again. This amounts to using the punched paper tape as an auxiliary backing store. If the
computer is equipped with magnetic tape it is better not to use the ordinary matrix scheme but to use
instead the Magnetic Tape Matrix Scheme (R 7502). This scheme is basically similar to the original
scheme but includes a few extra orders for transferring information between the main store and the
magnetic tape store. The scheme uses two magnetic tape mechanisms (see Chapter 10), on each of which about
200,000 locations are available for storing matrix elements (more can be made available by changing
tapes).

Occasionally the precision of the numbers represented in the matrix scheme may be inadequsite. If
the calculation is not too large it may be possible to use the Double Length Matrix Scheme (R 7503),
in which the precision is equivalent to about 22 decimal digits. All calculations are done in a
double-length floating-point mode in which each number is represented by three words. The conventions
for writing programmes are very similar to those for the ordinary matrix scheme except that only about
2000 natrix elements may be stored. Operations take roughly three times as long as those in the
A ordinary scheme.

V 9.5 Double-length floating-point arithmetic

For certain calculations high precision is required. The Double-Length Floating-Point Interpretive
Scheme, R 650. has been written for them. ^ The programme is written in the form of a subroutine, which
can be called in when it is required to do double-length floating-point operations; the user has
therefore to be familiar with Pegasus programming and with the use of the Assembly routine. The
interpreted orders resemble ordinary Pegasus orders and can consequently be read in, together with
them, by the Initial Orders. Entry to the interpretive scheme and exit from it are very quick and
simple; it is thus easy to do special counting operations and the like with ordinary machine orders.
This routine occupies only 32 blocks in the main store.

The numbers used with R 650 each occupy three words, the first two of these represent the double-
length normalized (or zero) argument and the third holds the binary exponent as an integer. There is
a floating-point accumulator which holds one number; this occupies Ul.l, 1.3 and 1.5. All the other
numbers are held in the main store and are referred to by decimal addresses, which are always multiples
of three. The address of a number is actually that of its first word (the left half of its argument),
counted from the start of the R 650 working space. This working space is a section of the main store
set aside to hold double-length floating-point numbers; there is a preset-parameter to fix the start
of the working space. The second and third words representing a number are stored in the same position
as the first word, but in the two following blocks.

Interpreted orders (i.e. orders causing R 650 to carry out operations on double-length floating-
point numbers) are always obeyed in UO. As usual they are obeyed sequentially until a jump occurs. If
the last order in UO (i.e. in 0.7+) does not cause a jump then the next order to be obeyed is always
in 0.0 (whether it is to be interpreted or not). Each interpreted order is made up of four parts
N X F M like an ordinary order; but often a minus sign is written in the X part and the single address
in the order is then denoted by 5. This is usually the address of a number and will therefore be a
multiple of three; the number itself is the content of 5 and is denoted by s. In explaining the
effects of the various orders the double-length floating-point accumulator will be denoted by A and its
content by a. For example, the order

12 - 00

has .S = 12 ( a multiple of 3), f = 00 and A/ = 0; its effect is to add the double-length floating-point
number from location 12 into the floating-point accumulator. The effect of a OO-order can in general
be written

a = a -^ s.

The order

150 - 44

t This programme was written by Mr. J. G. P. Francis while with the National Research Development Corporation.

- 219 -

9.5

SOME PROGRAMMING TECHNIQUES

writes the number in the floating-point accumulator into location 150. The effect of a 44-order may
be written

s - a.

The whole order-code of R 650 is summarized in Table 9.9. many of the details are explained later: the
times given in the last column are expressed in milliseconds; they are approximate only. Extravagant
effects may occur if any unassigned order is obeyed.

00
02
04
06

10
12
14
16

20
22
24
26

30
32
34
36

40
42
44
46

50
52
54
56

60
62
64
66

70
72
74
76

Operation

= a + s

= a + s, and set indicator if a'

= a + s (same as 00)

= a + s, and set indicator if a'

= a - s

= a - s, and set indicator if a

= s - a, and set indicator if a = s or -a

= N-th constant in list (N > 1)

= {.' = number read from input tape

punch CR LP and value of a

punch Sp Sp and value of a

a or s (same as 02)

5 indicates the style

test indicator: jump in to N if not set. Clear indicator
test indicator: jump out to N if not set. Clear indicator
count: x]^ = x^ +3. x^, = x^. - 1; jump in to N it x'q f

count:

- X,, +3, Xr

1; jump out to /V if jc' ?*

block-read and jump: Block Af(+) to [/O, jump to O.J
a' = 5

set indicator if exponent of a < exponent of s
set indicator if \a\ < 2

a = a X s

a' - integral part of a

a/ s

subroutine operation 5

Time
(msec)

51
51
51
51

24
26
26
27

38
38
38
38

34
26
30
24

48
29

Table 9.9 Double-length floating-point orders.

As an illustration, we give below a sequence of interpreted orders for reading in a number x and
placing in the floating-point accumulator the quantity

x(x + 3)/(x -3).

The sequence is as follows.

220

SOME PROGRAMMING TECHNIQUES 9-5

3-52 a' = 3

0-44 write 3 into location 50

3-22 X to A and 53

0-10 X - 3

6-44 X - 3 to 56

3-40 X to A

0-00 X + 3

3-60 x(x + 3)

6 - 70 x(x + 3)/(x ' 3) to A

It will be seen that all the functions included in Table 9,9 are even (i.e. they have an even
second digit). If the second function digit in any order is made odd by the addition of 1 then the
order is obeyed as usual but the next order is not interpreted; it is treated in the normal way as a
machine order. This is the simplest way of leaving R 650 and entering machine orders.

The indicator referred to in Table 9.9 can be used to facilitate convergence tests in iterative
processes; it is actually represented by the sign bit of U5.1, which is 1 if the indicator is set
(the rest of 5.7 is used for other purposes). In the jump orders, jump in means that R 650 is to
start interpreting orders at the specified address, and jump out means that the jump is to an ordinary
machine order. All the jumps can be only to orders in UQ; the block part of the jump address is not
used. The two orders for counting are of interest, they advance the modifier by 3 because of the way
numbers are addressed; there is no carry from x^ to x,^ (as in an ordinary 67-order) ; no jump occurs
if x„ overflows and a jump will always take place if JO is specified. The 50-order jumps in, and the
51-order jumps out. Machine orders may use UO and Ui and all the accumulators without disturbing the
interpretive routine.

Entry to the scheme is normally by obeying its cue 02. Before doing this we must place m J6 the
address of the first order to be interpreted (which must be in UO); for example to start interpreting
orders at 0. 1+ we obey the order

to. ly 6 40

and then obey cue 02 to R 650. If instead we use cue 01 then the specified order will be treated as an
ordinary machine order. If we leave the scheme to obey machine orders which do not disturb the
interpretive routine (in Ul, 2, 3 and 5) we can at any time re-enter it by jumping to 1.7+, after
placing in X6 the address of the next order to be interpreted. Thus to start interpreting at 0.3 we
can use the following pair of orders.

^0.3j 6 40
1.7+ 60

If we wish to start interpreting at 0.0 we can simply jump to 1.7 without setting X6. On exit from
interpreted orders XI contains the most-significant word of the floating-point accumulator (this may be
tested in the usual way by ordinary orders 60 to 63); and X2, 3, 4, 5 will be unchanged since entry,
except for the result of counting (or using an interpreted 20-order).

As an example of some of the techniques which may be used we give on page 222 a complete programme
using R 650 which reads from the tape fifty sets of numbers x, y, z, w and calculates and prints for
each set the quantity

-(X + y)/(zw + 2).

The data tape is to be placed in the main tape-reader when the E-directive causes a 77-stop.

Tlie output orders are 24 and 26; they print the number in the accumulator. 24 preceding it by
CR LP. and 26 by Sp Sp. The number 5 specifies the style. If b decimal digits are to be printed
before the point and c after it then 5 = 326 + c; for example 140 = 32 x 4 + 12. so that style 140
causes 4 digits to be printed before the point and 12 after it. If 6 = (i.e. 5 ^ 31) or if the
number is greater than about 10^2, then it is printed in standard floating decimal form; this has a
signed argument with the point after the first significant digit, and then c digits, a space, and a
signed decimal exponent. Not more than 21 or 22 significant digits are printed in the argument, and
the exponent is printed as zero if the argument is zero. The number is always correctly rounded and
left-hand zeros in the integral part are replaced by spaces. If the number is an exact integer then
the decimal point and the fractional part are replaced by spaces. During output 60 is used to store
the accumulators. (A number whose exponent exceeds 511 causes a 77-stop and then printing of * 0) .

The input order has the function 22; this order reads a single number from the input tape,
converts it to double-length floating-point form and places it in location 5 and the accumulator. The
number may be punched as an integer or a fraction or a mixed number and can. if desired, be followed
by a decimal exponent; both the argument and the exponent must be signed and the whole number is
always terminated by CR LP. Numbers punched by either of the output orders (24 or 26) are always
suitable for re-input provided each number is termined by CR LP. In front of the number CR LP. LP, Sp
and 4) are ignored and Er may, as usual, appear anywhere except between CR and LP. Spaces may be punched
anywhere except between digits or between the exponent and the terminal CR LP; there must be at least
one space between the argument and the sign of the exponent (if there is one). A single decimal point

221 -

9,5

SOME

PROGRAMMING

TECHNIQUES

may be punched anywhere In the argument and may precede or terminate it. Any number of digits may be
punched in the argument but only 21 or 22 significant digits are used, the rest being ignored. The
number zero is stored with an exponent -2^^. If a number is read whose decimal exponent is 128 or
larger in absolute value there is a 77-stop, but the programme will continue if the Run key is operated;
this has been devised to detect punching errors. During input the programme uses 50.

E.2.0

FIFTY NUMBERS

R
650

-0
01

0.0

T1+

140

1.7

0.6+

L
A2
A3
E2.0

Call for cue 02 to
R 650 (in 0+, 1)

prepare to interpret at 0.2
set counter

+ cue 02
to R 650

2 to 50

B1+ to m. jump to 0.0

from data tape

X to 53

y to 4 J

X + y

-(x + y) to 53

X to 56 1

w to A J

zw to A

zw + 2

zw + 2 to 56

-(X + y) to A

-(X + y)/(zw + 2)

print result (style 140), then obey machine orders

count in 2^, start interpreting at 0.0

Interpreted orders can be modified, but only by a modifier in X3, 4 or 5. When any order is
modified all thirteen bits of the modifier are added to the first ten bits of the order, i.e. to
5(or else to N and X) . which are extended by 3 bits (except for the 56-order); this modification
cannot cause overflow. Modification can be used in a loop with either of the count orders (34 and 36).
The counters and modifiers are normally set by a 20-order, which is described later.

222 -

SOME

PROGRAMMING

TECHNIQUES

9.5

The start of the working space used to hold numbers is fixed by a preset parameter; If no
parameter- list is supplied an optional list will set it to B125.1 (this is the largest address that
may be set). To set the start of the working space to, say, BllO.O the programmer must include the
following parameter- i is t in his programme.

-0

650

110

A number of subroutines, mostly for matrix work, are available for R 650 and it is sometimes useful
to use programmer's subroutines also. The use of all subroutines is greatly facilitated by the
special 76-order. If this order is to be used the programmer must supply a parameter-list, followed
immediately by an index of the subroutines (and cues) which are to be used. The preset -parameter in this
list is the one fixing the start of the working space, and is written as above except that its "fc-order"
is now the number of different subroutine operations. After this parameter we put a number of words
each specifying one subroutine operation; we write the routine -number in the "a-order" and the cue-
number in the "fc-order". Consider the following parameter-list and index, for example.

-0

650

120

680

>■ title

' Parameter-list

working space B120.0

two subroutine operations

R 680 cue 01
. R 680 cue 02

Index of subroutine
operations

The interpreted 76-order carries out the operation defined by the .S-th entry in the index;
example, if the above index is supplied then the order

for

2-76

causes R 680 to be called in by cue 02. When using subroutines in this way no link need be set; after
the subroutine operation is complete the next order is interpreted (the 77-order causes the following
order to be treated as an ordinary machine order when the subroutine operation is finished). The
parameter- list and index are treated in a special way by R 650; they must not be overwritten if
subroutines are to be used. They are actually processed by an interlude and do not contain what was
written there by the programmer.

The following are brief specifications of the available subroutines. Most of them require the
setting of certain modifiers and counters as programme-parameters; this can be done by using the
20-order, which is described below.

R 680 Linear combination

Store: 4 blocks.

Cues: 01 av replaces u (time 126 + 43n msec) ,

02 I! + av replaces u (time 126 + 58 n msec).

Before entry put 4jj^ = address of vector u, 5nf = address of vector v, 5^ = number of elements
in each vector.

R 681 Scal ar product

Cues: 01 a' = u.v (time 117 + 59n msec).

02 a' = a + u.v (time 131 + 59n msec)
Before entry set accumulators as for R 680.

R 68^ Square root

Cue: 01 a' = /a (time about 224 msec).

R 685 Matrix multiplication

Store: 3 blocks.

Cues: 01 B X C replaces D

02 B X C is added to D

(time about

60 pr(q + 3) msec)

Store: 3 blocks.

Store; 2 blocks + 3 more for R 681.

223

9.5 SOME PROGRAMMING TECHNIQUES

Before entry set (d, p) in 13, (c, r) in J4 and (b, q) in ^. The matrices B, C and D,
which are stored by rows, have their first elements at 6, c, d and dimensions p ^ q, <? x r and
p X r respectively. The matrix D must be different from B or C. The content of the floating-
point accumulator is lost. , This routine uses R 681 as a subroutine.

R 686 Matrix division

Cues: 01 B"^C replaces C, B* replaces B. Store: 5 blocks + 9 more for R 680,

681, 685.

02 C, B~^ replaces C, B* replaces B.

03 B'^C replaces C, where B* is given.

04 C.B"i replaces C, where B* is given.

Time: 60pq(p + 3) msec for cues 03 and 04; for cues 01 or 02 add p^(29p + 240) msec, to this.

Before entry set (c, q) in J4 and (6, p) in A'5. The matrices B and C, which are stored by rows,
have their first elements at 6 and c; B has dimensions p x p and C has dimensions p x q (cues
01 and 03) or q x p (cues 02 and 04). A special method of division is used in which B is
replaced by B*. the basic information needed to reduce it to the unit matrix by straightforward
elimination above and below the diagonal (no search is made for pivots); this process requires
about % p3 arithmetical operations. This information 8* is all that is required to divide
ajiother matrix by B, and operating with it resembles a matrix multiplication in needing only
about p^q arithmetical operations. Once B* has been found by use of cue 01 or 02 it can be used
again by cue 03 or 04 (for example to find B'^C). The content, of the floating-point accumulator
is lost. This routine uses R 680 and R 685 (which uses R 681).

A subroutine for Power Series Economisation is also available for R 650; details of this
may be found in the library specification, R 360.

The 76-order automatically restores the whole of the computing store, except for Jl, 6, 7 and the
floating-point accumulator A, after a subroutine operation is complete. It is qu-ite possible to have
subroutines within subroutines (i.e. sub-subroutines), and so on; each 76-order proceeding to a lower
level causes R 650 to use extra blocks to store certain information temporarily. Blocks 894 and 893 are
used with a simple subroutine; if this calls in its own subroutine JS892 and 891 are also used, and so
on.

Programmer's subroutines may also be used with R 650; they should be written in the usual way for
Assembly. They may consist of machine orders only, or partly of interpreted orders (the latter must, as
always, be in UO). The cues must be machine orders, for example

1.7 60

will enter the subroutine at O+.O as an interpreted order (see above). Blocks Ul, 2, 3 and 5 may not
be disturbed if interpreted orders are to be used later, unless they are restored by a cue to R 650.
Note that f/1.1, 1.3 and 1.5 are the floating-point accumulator. During the process of entering a
subroutine XI, 6 and 7 are used but the rest of the computing store is unaltered (except for the block
overwritten by the action of the cue); however XI will not contain the most-significant word in the
accumulator. When the subroutine has finished its work it should obey cue 05 to R 650, which is called
for by a, tag in the usual way. When this cue is obeyed the whole of the computing store (except XI, 6
7 and the accumulator A) will be restored to its condition on entry to the subroutine. Ordinary
subroutines may be used by machine orders within programmer's subroutines in the usual way; but if
R 650-subroutines are to be called in by interpreted 76-orders a special technique must be adopted. If
the sub-subroutine is included in the index of subi'outlne operations then the usual 76-order will call it in;
it may, however, be undesirable to have to draw up the index at the time when the programmer' s subroutine
is being written. If this is so we must arrange that UO.l contains a cue to the sub-subroutine at the
moment it is to be entered; the special order - 76 is then used to enter it. This technique ensures
that correct action is taken about the level of the subroutine.

The interpreted 20~order can be used for setting modifiers and counters (or other constants) in
X3, 4 or 5. If this order is to be used the programmer must supply a special list of constants; this
always follows immediately after the working-space address parameter and the subroutine index (if there
is one). For example, the order

2 5 20

puts the second constant in the list into 15 (the first constant is numbered 1). This order is a very
useful one, as will be seen from the next example.

Suppose we are given three square matrices A, B and C, each with 6 rows and columns, and a 6-vector
u and we have to evaluate the vector

A-i(A-iB + C)-%

and print it with three digits before the point and five after. We assume that the matrices are punched
by rows on a data tape in the order A, B, C and are followed by u. There are altogether 114 numbers on
this tape; these are read into consecutive locations starting at 50. The matrix A occupies 50 to 5105
(since every address is a multiple of 3), the matrix B occupies 5108 to 5213, C occupies 5214 to 5321 and
the vector u is placed in 5324 to 5339.

- 224

SOME PROGRAMMING TECHNIQUES

9.5

E 2.0

D
N

VECTOR V
Al

R
650

-0
01

0.0

.©

—

0.2

—
52

(0, 114) to X3

prepare to interpret 0.2

+ cue 02 to R650

. read in A, B, C, u

(b, 6) to X4

(a. 6) to ^5

A" % replaces B, A* to A

(c. 36) to J5

1 to accumulator

Add C to A"%

(u, 6) to Xi

(b. 6) to X5

(A" % + C) ~ ^u in place of u

0.0

C5i

0.1

0.2-^

(a, 6) to X5

a"Va'% + C)-%
I. print the answer

225 -

9.5

SOME

PROGRAMMING TECHNIQUES

-0

650

100

686

680

686

108

-0

-D

216

-0

324

-0

Title

working space address 100.0
no. of entries in index

Parameter
list

y subroutine index

* Constant list

(108, 6)

(0, 6)

(216. 36)

(324, 6)

L
A2
A3
E 2.0

The tape for R 650 is not included in the Library tape but is placed in the second tape -reader
when required. It contains R 650 itself, followed by a special 4-block interlude and the subroutines.
The interlude tampers with the parameter-list in a special way and ensures that the appropriate
subroutines are accepted; in addition the optional parameter- list is replaced (if necessary) by two
words containing the working-space address in the a-order and 6-order respectively, and the programmer' s
parameter is overwritten by information concerned with subroutine entries. The parameter-list
actually used by Assembly is therefore always in the place of the optional list. If the user needs the
address of the working space he should call for R 650 parameter 01 or 02. The address of R 650 itself
may be obtained by calling for cues 03 and 04; these are ordinary partial cues containing the address
in the /V-part of the a-order and the fc-order respectively. While R 650 and its subroutines are being
read in BO.O and 0.2 are used.

When interpreted orders are being obeyed t/5.4n| normally holds 3 and the rest of the word is clear
(thus C(5.4) = 3,2"^^); this constant is added to the specified accumulator by the count orders 34 and
36. The user can change this word to another value; for example, he can set it negatively for working
backwards through the store, or perhaps to some multiple of 3.2"^^ for referring to a column or the
diagonal of a matrix. It is however used by the subroutines doing matrix and vector operations and
must normally contain 3.2"^"' before the 76-order calls In the subroutine. The subroutine R 680 may be
used if the elements of the vectors have addresses differing by a constant difference other than 3;
this difference should be set in 5.4j)/; similarly in R 681 the vector u (but not v) may be so handled.
In any event C(5.4) is always reset to 3.2"^"^ after all 76-orders.

No provision is made for the possibility of any exponent overflowing. The overflow-indicator OVR
is always cleared by R 650 on entry and will be left clear on exit. The order 0-52 sets zero in the
accumulator with an exponent -2^'; this kind of zero is also stored if a 22-order reads zero from the
tape. Zero resulting from the subtraction of equal numbers has an exponent 76 less than that of the
numbers. The exponent of zero resulting from multiplication or division is the sum or difference,
respectively, of the exponents of the operands.

If we make the order-pair in B23+. 2 of R 650 a stop order-pair then an optional stop will occur
before each interpreted order; at this instant X6 will contain the order-pair and 7j^ will contain its
address (and a negative sign-bit for an a-order). On entry initially, this stop will occur once before
X6 and 7 have been set. In general, if the programme stops while interpreting an order then ULG^f will
contain the address of the next order. Stops are liable to occur on input or output (for punching
errors or numbers with very large exponents), or on dividing by zero, or if a 20- or 76-order is
interpreted and no parameter has been supplied.

226 -

Chapter lo
Magnetic Tape

A basic Pegasus installation can be augmented in various ways and the ancillary equipment is so
designed that it is still possible to run, without change, programmes prepared for the basic installation.
In this chapter we describe magnetic tape equipment and its associated programming techniques.

10. 1 General description of magnetic tape equipment

Magnetic tape equipment can be added at any time to a basic Pegasus installation; it provides an
auxiliary store of very large capacity. The equipment can also be used with the converter fitted at
some installations, when the tape can be regarded as a medium for input and output. The magnetic
tape itself is ^2 inch wide (13 mm) and is normally wound on metal spools having a diameter of 9% inches
(24.8 cm). The tape on a spool is called a reel; it is normally 3000 feet long (914 metres), but
lengths of 600 feet (182 m. ) and 1800 feet (549 m. ) are also used. The tape is made of thin, flexible,
inextensible plastic material coated on one side with a magnetic iron oxide; information can be recorded
(or written) on to it and read from it by electromagnetic heads in much the same way as with a magnetic
drum. Any information written on the tape will remain there indefinitely and can be read as many times
as desired until it is overwritten by new information. The process of writing on to a part of the tape
a,utomatically erases anything previously written there, whereas the reading process does not disturb the
recorded information. Each reel of tape is permanently marked out into sections, on each of which either
16 or 32 words can be recorded, depending on the way the tape was originally marked out. The sections
marked on a reel of tape are all either of 16 words or of 32 words; no mixture of section- lengths is
possible on a reel. Each sedtion has an address, which is used to identify it and is permanently
recorded on the tape. This address is simply an integer; the first section on any reel always has the
address 0, the next 1, and so on. The number of sections on a reel depends on the exact length of the
reel, on certain physical properties of the coating, and on the section-length. A 3000 ft. reel has
about 10976 16-word sections (a total of 175616 words), or alternatively about 8416 32-word sections
(269312 words). The storage capacity of a single reel of tape is thus very large - about thirty times
that of the main store of the computer (or the equivalent of about 10976 or 16832 80-column punched cards).

When a reel of tape is to be used with the computer (or the converter) it is mounted (or loaded) on
a tape mechanism, which incorporates the reading and writing heads and equipment for moving the tape at
high speed in either direction and for spooling it up. It is quick and easy to load a reel on to a
mechanism or to unload it; in fact it is usual to allow I'/z minutes to change a reel. Each mechanism
has a small control panel; one of the switches on this panel must be set to show whether a 16-word or a
32- word tape is in use on the mechanism.

There are several tape mechanisms in an installation; and also one or two tape control units, each
of which contains the electronic circuits for controlling up to five mechanisms, and for reading, writing,
checking and storing information. The tape mechanisms are numbered from to 4 (if there are five), in
the order determined by a plugboard on the tape control unit which may be set up by the programmer. The
arrangement of the plugboard should be altered only when the control unit is not obeying a tape operation, and
when the Run key on the computer is in the STOP position. Each tape control unit has its own control
panel, which is chiefly of concern to the maintenance engineers. On the right of this control panel is
a set of five keys which may be used by the programmer to isolate any of the mechanisms, i. e. to prevent
writing, thus safeguarding the information on a tape. If a key is in its normal position (tape write)
information can be written on to a tape on the corresponding mechanism; if the key is depressed (write
inhibit) information cannot be written. These keys provide complete protection from programming errors
and faults.

10. 2 Programming with magnetic tape equipment

From the programming point of view the tape control unit is capable of obeying tape orders as
described below; it contains a buffer store, which can be regarded as a waiting room between the magnetic
tapes and the computing store.

When magnetic tape equipment is fitted to the computer two additional special registers with
addresses 20 and 21 are provided; these are described below. Also provided when magnetic tape, punched
cards or a line-printer are used, is a buffer transfer order (76) in which the yv-digits specify a function
as well as an address.

The magnetic tape buffer store can hold 32 words divided into four blocks WO, Wl. W2 and If3. The
76-order with an ^-address between and 23 is used to transfer blocks of 8 words between this buffer
store and the computing store, as shown in the following table; information in brackets is relevant only
to an installation in which there are two tape control units and therefore more than five tape mechanisms,
and an extra buffer store, M, W5, W6 and Wl.

227

10,2 MAGNETIC TAPE

N- address 76- order

0-3 Interchange any buffer block of tape

control 1 with any computing store block.

r 4 - 7 Interchange any buffer block of tape 1
L control 2 with any computing store block. J

8-11 Transfer any buffer block of tape control 1
to any computing store block.

r 12 - 15 Transfer any buffer block of tape control 2 "1
L to any computing store block. J

16 - 19 Transfer any computing store block to any
buffer block of tape control 1.

20 - 23 Transfer any computing store block to any 1
buffer block of tape control 2, J

The 76-order with an iV- address greater than 23 is described in Chapter 11. t

The 76-order is written in a similar way to a 72- or 73-order, the first address specifies the

function and, modulo 4, the buffer store block, and the second address specifies the computing store

block. For example, the order

2 [5]76

interchanges the contents of U5 in the computing store and W2 in the buffer store, and the order

16 [i]76

writes the contents of f/3 in the computing store to m in the buffer store. Pour of these orders are
needed to change all the words in the buffer store. When using a 32-word tape the whole content of the
buffer store is written on to a section of the tape or is replaced by a section read from the tape.
When using a 16-word tape either half of the buffer store (i.e. either WO and Wl or W2 and If3) is
concerned and the other half is not affected; we can then consider the buffer store as containing two
sections of information which are referred to as section (ffO and Wl) and section 2 (W2 and ff3).
Pig. 10. 1 shows the buffer store and its relationship with the other stores.

A tape order is initiated by sending a word to the special register 20, for example by an order
such as

20 5 10

The word sent to register 20 is called a tape control word; it determines the kind of operation to be
carried out. In a large installation with two tape control units, a tape order referring to a mechani
linked to the second tape control unit is initiated by sending the tape control word to special
register 21; register 20 refers only to tape control unit 1. In what follows we shall assume that
only one control unit is available.

The tape control word may be written on a programme sheet as a pseudo order-pair. A general tape
control word may be written thus:

fg h

Here A is the address of the section on the tape,

/ is the number of the tape mechanism (0, 1, 2, 3 or 4) ,

g specifies the section of the buffer (0 or 2),

h indicates the kind of operation to be done, according to the following code;

h - Search,

h = 1 Read,

/i = 2 Write,

/i = 3 Rewind.

More details are given below, but we first give an example. Suppose that a 16-word tape has been loaded
on to mechanism No. 1 and we wish to write on to section 5432 the contents of (74 and (75 in the computing
store. The following sequence of orders may be used.

t In Pegasus 1 with magnetic tape, the 76-order is available only as an interchange order, that is with
an N-address 0-3; an N-address greater than 3 in interpreted modulo 4.

228 -

MAGNETIC TAPE

10.2

Accumulators
(X)

Special
Registers

Ordinary
Registers

OVR

Computing
Store

1 „

2 _

3 _

4 „

5 _

6 _

7 _

__„_'

yui

1.0

1 .

2.
3

4
5
6.

7 .

2.0

1 .

2 .

3 .

4 ,
5.
6.
7.

3.0 .
1

4.0
1

2
3

5,
6

5.0
1
2
3

4
5
6

□

Punched paper tape
input and output

Tape readers

mam

second

xz:-'

output
punch

EXTERNAL

CONDITIONING

RELAYS

Checking equipment

>U2

\U3

L[/4

Buffer
transfers:
any block
in com-
puting
store and
any block
in buffer
store

Block
transfers

Magn

WO .
Wl .
W2 .

etic tape cc
and Buffer

)ntrol
Store

a
o

•H
4->

o
a:>

> m

-o
u

1
CO

■r-t
-P
O

► w

1
CD

r-(

unit

c
o

■rH

> -a

(M
CO

W3 .

Magnetic Tape
Mechanisms

Reading
or

writing

-*<

Any whole
Section

(^~^

(^"^

Isolated:
no writing
possible.

Main Store

BO .

Bl .

B895

B896

B1023 .

Pig. 10. 1 Diagram of Pegasus and magnetic tape.

229

10.2

MAGNETIC TAPE

Start

0.0

5432

12 2

18
19

C1176
[l76

0.0
20

4 00
4 10

A = 5432. / = 1
g = 2. h= 2
transfer (74 to W2
transfer (75 to W3
tape control word to Xi
initiate tape order

. tape control word

In the tape control word, which is written for convenience in 0, 0, the address is 5432, / = 1
specifies the mechanism to be used, g = 2 means that the transfer to or from tape concerns section 2 of
the buffer store (W2 and If3), and h = 2 shows that we wish to write on to the tape. The computer first
obeys two 76-orders, which put the contents of f/4 and f/5 into W2 and W3 respectively; the original
contents of W2 and W3 will be lost. ♦ The next two orders send the tape control word to register 20 to
initiate the writing operation: the magnetic tape control unit will promptly start searching the tape
on mechanism No. 1 for section 5432 and will write the information on that section as soon £is it has been
found (without disturbing the contents of the buffer store). As soon as the tape order has been
initiated by sending the tape control word to register 20 the rest of the process is carried out quite
automatically by the tape control unit; the computer is free to carry on with the rest of the programme.
The computer will be made to wait if it encounters an order referring to register 20 before the tape
operation is complete. This will also happen if a 76-order is found referring to W2 or H'3; but the
other section of the buffer store may be freely used. Complete disregard of all such considerations
cannot cause faulty operation. Magnetic tape operations are autonomous since they are carried out while
the computer is doing other operations.

The kind of operation which is carried out by the tape control unit depends on the quantity h in the
tape control word. The four operations are as follows.

/i = (search). Move the tape (forwards or backwards) to the prescribed address, then await a

further tape order,
/i. = 1 (read). Read the information in the specified section of the tape and copy it into the

buffer store,
/i. = 2 (write). Write the information in the buffer store on to the specified section of the tape.
/! = 3 (rewind). Rewind the tape on the specified mechanism.

The search order may be used when we know that the section we require is some way along the tape; to be
able to use it we have to anticipate which section we shall need next, and also be able to carry on with
other operations while searching. The read and write orders include a preliminary search for the
section required, and we are therefore not obliged to precede them by a search order. The tape control
unit is capable of carrying out only one tape order at a time; it cannot, for example, be searching the
tape on one mechanism while writing on another. If we initiate one tape order before the previous one
is complete then the computer is held up. The rewind operation is, however, quite autonomous; as soon
as rewinding has been initiated on one mechanism then the tape control is free to operate with the other
mechanisms while the first tape is being rewound. Note that g is the part of the tape control word
used to indicate which section of the buffer store is to be used in 16-word reading and writing
operations; the value of g is immaterial when searching or rewinding or when we are using a 32-word
tape.

The binary digits in the tape control word are allocated as follows:

A f g h

t' S/' ' ■>/ *■ N

1 2

3 to 16

17 to 29

30 31 32

37 38

unused

address of
section

unused

mechanism

section of
buffer

unused

The values of the unused digits are entirely without effect on the tape control unit. The address in
the tape control word is an Integer with its least-significant digit in digit-position 16; if we have a
tape control word in an accumulator and we add 2~^^ to it the word will have its address increased by

f Provided that it is not necessary to keep the contents of t/4 and 1/5 in the computing store, the 76-
orders could be written as interchange orders:

2 Bl 76 interchange W2 and U4

3 m 76 interchange W3 and U5

When these orders are obeyed, the contents of Ui and 1/5 will be transferred into W2 and H'3 and the original
contents of W2 and W3 will be copied into t/4 and t/5. These orders would normally be written in this form
for Pegasus 1, as the 76-order is available only with an /V-address between and 3. If an yv-address
greater than 3 is encountered on Pegasus 1, the 76-order will be obeyed as an interchange order i.e. the
/V-address will be interpreted modulo 4.

230

MAGNETIC TAPE

10.2

one. This is a common requirement in magnetic tape programmes;
provided in a special register, whose address is 36. The order

the constant 2'

has therefore been

36 5 01

will increase by one the address part of a tape control word in X5.
used to change the other digits; for example, the order

The orders of group 4 can often be

5 41

will add one to h and can therefore be used to turn a read word into the corresponding write word.

Register 20 is made up of a nickel delay line in the tape control unit. When an order of group 1
is obeyed by the computer the final content of this register determines the tape order to be carried out.
Other orders may refer to it without initiating a tape order; for example, the order

20 5 00

will place in X5 the tape control word which Initiated the last tape order, and will have no effect on
the tape control unit. An order like this will hold up the computer if the last tape order is not
complete. Orders like the following are sometimes useful.

36 3 00

20 3 11

These cause the last tape order to be repeated, but with its address increased by one. They are thus
equivalent to an order for reading the next section, if the last tape order was a read order.

Programmes which write information on to magnetic tape should not use a tape address below section 1,
as it is customary to record in section special identifying information about the tape. The use of
section is described further on Page 234.

As an example of a short sequence using magnetic tape, let us suppose that we have a programme
recorded on 27 32-word sections on tape mechanism 2 starting at section 13. This programme is to be put
into the main store in blocks J52 to B109, after which B2 is to be copied into M) and a jump to 0.0 is to
take place. The following sequence will do this in 2.3 seconds (if section of the tape is under the
reading head initially); of this time 0.7 seconds is spent searching for section 13. Note that the
block transfers to the main store are done while the tape control unit is reading the next section.

0.0

1.0

13 - --

20 1

00 0,

0.0 5 00
20 5 10

^ 4 40
36 5 01

8 ©76

9 {176

10 S]76
n [1176

20 5 10
2 [173 4

3 H 73 4

4 JU 73 4

Tape control word for reading
section 13 on mechanism 2

(4.0, 0)

Tape control word to X5

Read section 13

(0, 27) to J4

Increase address in tape control word

All 32 words in
^ buffer store to
U2, 3, 4, 5

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